🧠:<think>Okay, so I have this problem about a chef who wants to optimize honey production from bees in an olive grove. There are two parts to this problem. Let me start with the first one.1. The olive grove has an area of ( A ) square meters. The number of bees ( B ) that can be sustained is directly proportional to ( A ), so ( B = kA ). They mention that the optimal number of bees per square meter is ( frac{n}{m} ). I need to find the relationship between ( k ), ( n ), and ( m ).Hmm, so if ( B = kA ), then the number of bees per square meter would be ( frac{B}{A} = k ). But they say the optimal number is ( frac{n}{m} ). So does that mean ( k = frac{n}{m} )?Wait, let me think again. If ( B = kA ), then ( frac{B}{A} = k ). So, ( k ) is the number of bees per square meter. Therefore, if the optimal number is ( frac{n}{m} ), then ( k = frac{n}{m} ). So the relationship is ( k = frac{n}{m} ). That seems straightforward.2. The honey production ( H ) is given by ( H(B) = alpha B - beta B^2 ). The chef needs exactly ( P ) kilograms of honey. I need to find the number of bees ( B ) that should be sustained to meet this requirement. Then, using the relationship from part 1, find the area ( A ) required.Alright, so ( H(B) = alpha B - beta B^2 = P ). So we have a quadratic equation in terms of ( B ):( -beta B^2 + alpha B - P = 0 )Let me write it as:( beta B^2 - alpha B + P = 0 )Wait, actually, multiplying both sides by -1:( beta B^2 - alpha B + P = 0 )So, quadratic equation: ( beta B^2 - alpha B + P = 0 )To solve for ( B ), we can use the quadratic formula:( B = frac{alpha pm sqrt{alpha^2 - 4 beta P}}{2 beta} )But since ( B ) represents the number of bees, it must be a positive real number. So, the discriminant must be non-negative:( alpha^2 - 4 beta P geq 0 )Therefore, ( P leq frac{alpha^2}{4 beta} ). That makes sense because the maximum honey production occurs at ( B = frac{alpha}{2 beta} ), which gives ( H = frac{alpha^2}{4 beta} ). So, if ( P ) is more than that, it's impossible to meet the requirement.Assuming ( P leq frac{alpha^2}{4 beta} ), we can proceed.So, the solutions are:( B = frac{alpha pm sqrt{alpha^2 - 4 beta P}}{2 beta} )But since ( B ) must be positive, both solutions could be positive because ( alpha ) and ( beta ) are positive constants. However, in the context of bee population, we might have two possible numbers of bees that can produce the same amount of honey. But in reality, there might be an optimal number that's either the maximum or a specific point.Wait, actually, the function ( H(B) = alpha B - beta B^2 ) is a downward opening parabola. So, it has a maximum at ( B = frac{alpha}{2 beta} ). So, for any ( P ) less than the maximum, there are two values of ( B ) that give the same ( H ): one on the increasing side and one on the decreasing side of the parabola.But in the context of the problem, the chef wants to meet the demand while maintaining the health and productivity of the olive trees. So, perhaps he wants to choose the number of bees that is optimal, meaning the one that doesn't overstrain the grove. So, maybe he should choose the smaller number of bees? Or is there another consideration?Wait, actually, the number of bees is directly proportional to the area. So, if he uses fewer bees, he can have a smaller area, but he needs to meet the honey requirement. Alternatively, if he uses more bees, he might need a larger area, but perhaps that's not optimal.Wait, no. The number of bees is directly proportional to the area, so if he uses more bees, he needs a larger area. But the chef might want to minimize the area used or maximize the honey per area. Hmm, the problem says "to meet the demands of his restaurant while maintaining the health and productivity of the olive trees." So, maybe he wants to use the optimal number of bees per square meter, which was given in part 1.Wait, in part 1, the optimal number per square meter is ( frac{n}{m} ), so ( k = frac{n}{m} ). So, if he uses ( k = frac{n}{m} ), then the number of bees is ( B = frac{n}{m} A ). So, perhaps he should use the number of bees that's optimal per area, meaning he should choose the number of bees such that ( B = frac{n}{m} A ).But in part 2, he needs exactly ( P ) kilograms of honey. So, he needs to set ( H(B) = P ). So, perhaps he can choose either of the two possible ( B ) values, but considering the optimal density from part 1, he should choose the one that corresponds to the optimal density.Wait, maybe I need to think differently. Since ( B = kA ) and ( k = frac{n}{m} ), then ( A = frac{m}{n} B ). So, substituting into the honey production equation:( H(B) = alpha B - beta B^2 = P )So, solving for ( B ), we get the quadratic equation as before.But perhaps the optimal number of bees is when the density is ( frac{n}{m} ), so ( B = frac{n}{m} A ). But if we need ( H(B) = P ), then we can express ( A ) in terms of ( B ), and then substitute into ( H(B) ).Wait, maybe let's express ( A ) from part 1: ( A = frac{B}{k} = frac{m}{n} B ). So, substituting into ( H(B) ):( H(B) = alpha B - beta B^2 = P )So, regardless of the area, the honey production is a function of ( B ). So, to get ( P ), we need to solve for ( B ) as above.But then, once we have ( B ), we can find ( A = frac{m}{n} B ).Wait, so perhaps the steps are:1. Solve ( alpha B - beta B^2 = P ) for ( B ).2. Then, compute ( A = frac{m}{n} B ).But the problem says "determine the number of bees ( B ) that should be sustained... to meet this requirement. Additionally, find the area ( A ) of the olive grove required to sustain this optimal number of bees, using the relationship derived in the first sub-problem."So, perhaps the optimal number of bees is the one that gives the maximum honey production, but in this case, the chef needs exactly ( P ). So, if ( P ) is less than the maximum, there are two possible ( B ) values. But since the chef wants to optimize production while maintaining the health of the trees, he might choose the smaller ( B ) to avoid overusing the grove. Alternatively, he might choose the larger ( B ) if he has enough area.But the problem doesn't specify whether he wants to minimize the area or not. It just says to determine ( B ) to meet the requirement and then find the area required.So, perhaps both solutions are acceptable, but we need to express both possibilities.But let me check the quadratic equation again:( beta B^2 - alpha B + P = 0 )Solutions:( B = frac{alpha pm sqrt{alpha^2 - 4 beta P}}{2 beta} )So, two possible values for ( B ). Since ( B ) must be positive, both solutions are positive because ( alpha ) and ( beta ) are positive, and the discriminant is non-negative.So, the chef can choose either of the two numbers of bees to produce ( P ) kilograms of honey. However, considering the optimal density from part 1, he might prefer the number of bees that corresponds to the optimal density, which is ( frac{n}{m} ) per square meter. So, if he chooses ( B = frac{n}{m} A ), then he can compute ( A ) accordingly.But I think the problem is expecting us to express ( B ) in terms of ( P ), ( alpha ), and ( beta ), and then express ( A ) in terms of ( B ), ( n ), and ( m ).So, perhaps the answer is:Number of bees ( B = frac{alpha pm sqrt{alpha^2 - 4 beta P}}{2 beta} )And the area ( A = frac{m}{n} B )But let me think again. Maybe the optimal number of bees is the one that gives the maximum honey production, but in this case, the chef needs exactly ( P ), which is less than the maximum. So, perhaps he can choose either of the two ( B ) values, but to maintain the health of the trees, he should choose the smaller ( B ), which would require a smaller area.Alternatively, maybe the optimal number of bees is the one that gives the maximum honey per area. Wait, but the honey production is given as a function of ( B ), not per area. So, perhaps the optimal number of bees is the one that gives the maximum ( H ), but since the chef needs exactly ( P ), which is less than the maximum, he can choose either of the two ( B ) values.But the problem says "to meet this requirement" without specifying any additional constraints, so perhaps both solutions are acceptable. However, in the context of maintaining the health of the olive trees, it's likely that the chef would prefer the smaller number of bees to avoid overharvesting or overpopulating the grove. So, maybe he should choose the smaller ( B ).But the problem doesn't specify, so perhaps we just need to provide both solutions.Wait, but the problem says "determine the number of bees ( B ) that should be sustained". The word "should" implies that there is an optimal choice. So, perhaps the optimal number of bees is the one that maximizes honey production per area or something else.Wait, let's think about the honey production per area. Since ( H = alpha B - beta B^2 ) and ( B = frac{n}{m} A ), then ( H = alpha frac{n}{m} A - beta left( frac{n}{m} A right)^2 ). So, ( H ) is a function of ( A ). To maximize ( H ) per area, we can take the derivative of ( H ) with respect to ( A ) and set it to zero.But wait, the chef needs exactly ( P ) kilograms, not necessarily the maximum. So, perhaps he just needs to solve for ( B ) such that ( H(B) = P ), and then find ( A ) accordingly.So, in conclusion, the number of bees is given by the quadratic solution, and the area is ( A = frac{m}{n} B ).But let me write it step by step.First, solve for ( B ):( alpha B - beta B^2 = P )Rearranged:( beta B^2 - alpha B + P = 0 )Using quadratic formula:( B = frac{alpha pm sqrt{alpha^2 - 4 beta P}}{2 beta} )So, two possible solutions:( B_1 = frac{alpha + sqrt{alpha^2 - 4 beta P}}{2 beta} )( B_2 = frac{alpha - sqrt{alpha^2 - 4 beta P}}{2 beta} )Both are positive since ( alpha ) and ( beta ) are positive, and ( sqrt{alpha^2 - 4 beta P} < alpha ) because ( P < frac{alpha^2}{4 beta} ).So, both ( B_1 ) and ( B_2 ) are positive.Now, to find the area ( A ), we use ( A = frac{m}{n} B ).So, for each ( B ), we have:( A_1 = frac{m}{n} B_1 )( A_2 = frac{m}{n} B_2 )Therefore, the chef can choose either ( B_1 ) and ( A_1 ) or ( B_2 ) and ( A_2 ) to meet the requirement of ( P ) kilograms of honey.But since the problem mentions "maintaining the health and productivity of the olive trees", it's likely that the chef would prefer the smaller number of bees, which would correspond to ( B_2 ) and ( A_2 ), as it's more sustainable.However, without explicit instructions, perhaps both solutions are acceptable. But in an optimization context, sometimes the minimal solution is preferred to minimize resources.Alternatively, maybe the optimal number of bees is the one that gives the maximum honey per area, but that would require a different approach.Wait, let's think about maximizing ( H ) per area. Since ( H = alpha B - beta B^2 ) and ( B = frac{n}{m} A ), then ( H = alpha frac{n}{m} A - beta left( frac{n}{m} A right)^2 ). So, ( H ) as a function of ( A ) is:( H(A) = left( frac{alpha n}{m} right) A - left( frac{beta n^2}{m^2} right) A^2 )To find the maximum ( H ), take derivative with respect to ( A ):( H'(A) = frac{alpha n}{m} - 2 frac{beta n^2}{m^2} A )Set to zero:( frac{alpha n}{m} - 2 frac{beta n^2}{m^2} A = 0 )Solving for ( A ):( frac{alpha n}{m} = 2 frac{beta n^2}{m^2} A )Multiply both sides by ( m^2 ):( alpha n m = 2 beta n^2 A )Divide both sides by ( 2 beta n ):( A = frac{alpha m}{2 beta n} )Then, ( B = frac{n}{m} A = frac{n}{m} cdot frac{alpha m}{2 beta n} = frac{alpha}{2 beta} )So, the maximum honey production occurs when ( B = frac{alpha}{2 beta} ), which is the same as the vertex of the parabola in the original ( H(B) ) function.But in our problem, the chef needs exactly ( P ) kilograms, which is less than the maximum ( frac{alpha^2}{4 beta} ). So, he can choose either ( B_1 ) or ( B_2 ). If he wants to use the minimal area, he would choose ( B_2 ), which is smaller, leading to a smaller ( A ). If he wants to use the larger area, he can choose ( B_1 ).But the problem says "to meet the demands... while maintaining the health and productivity of the olive trees." So, maintaining health might imply not overpopulating, so choosing the smaller ( B ) and smaller ( A ) would be better.Therefore, perhaps the optimal ( B ) is ( B_2 = frac{alpha - sqrt{alpha^2 - 4 beta P}}{2 beta} ), and the corresponding area is ( A = frac{m}{n} B_2 ).But the problem doesn't specify whether to choose the smaller or larger ( B ), so maybe both are acceptable. However, in the context of optimization and sustainability, the smaller ( B ) is likely the preferred solution.So, to summarize:1. The relationship between ( k ), ( n ), and ( m ) is ( k = frac{n}{m} ).2. The number of bees ( B ) is ( frac{alpha pm sqrt{alpha^2 - 4 beta P}}{2 beta} ), and the area ( A ) is ( frac{m}{n} B ).But since the problem asks for the number of bees that should be sustained, implying a specific value, perhaps we need to choose the smaller ( B ) for sustainability. So, ( B = frac{alpha - sqrt{alpha^2 - 4 beta P}}{2 beta} ), and ( A = frac{m}{n} cdot frac{alpha - sqrt{alpha^2 - 4 beta P}}{2 beta} ).Alternatively, maybe both solutions are acceptable, but the problem might expect both possibilities.Wait, let me check the problem statement again:"2. The honey production ( H ) (in kilograms) from the bees in the olive grove is given by the function ( H(B) = alpha B - beta B^2 ), where ( alpha ) and ( beta ) are positive constants. Given that the chef needs exactly ( P ) kilograms of honey for his restaurant, determine the number of bees ( B ) that should be sustained in the olive grove to meet this requirement. Additionally, find the area ( A ) of the olive grove required to sustain this optimal number of bees, using the relationship derived in the first sub-problem."So, it says "the number of bees ( B ) that should be sustained", which suggests a single answer. So, perhaps the optimal number is the one that maximizes honey production per area, which we found earlier as ( B = frac{alpha}{2 beta} ). But wait, that's the maximum honey production, which is ( frac{alpha^2}{4 beta} ). If ( P ) is less than that, then the chef can choose either side of the parabola.But the problem says "to meet this requirement", so he needs exactly ( P ). So, perhaps both solutions are acceptable, but the problem expects us to provide both.Alternatively, maybe the optimal number of bees is the one that gives the required ( P ) with the minimal area, which would be the smaller ( B ).But since the problem doesn't specify, maybe we just need to provide the general solution.So, in conclusion:1. ( k = frac{n}{m} )2. ( B = frac{alpha pm sqrt{alpha^2 - 4 beta P}}{2 beta} ), and ( A = frac{m}{n} B )But to write it neatly, perhaps express ( B ) as ( frac{alpha pm sqrt{alpha^2 - 4 beta P}}{2 beta} ), and then ( A = frac{m}{n} cdot frac{alpha pm sqrt{alpha^2 - 4 beta P}}{2 beta} )Alternatively, factor out the constants:( A = frac{m alpha}{2 n beta} pm frac{m}{2 n beta} sqrt{alpha^2 - 4 beta P} )But perhaps it's better to leave it as ( A = frac{m}{n} B ), where ( B ) is as above.So, to wrap up:1. ( k = frac{n}{m} )2. ( B = frac{alpha pm sqrt{alpha^2 - 4 beta P}}{2 beta} ), and ( A = frac{m}{n} B )But since the problem says "the number of bees that should be sustained", maybe it's expecting the positive solution, but both are positive. So, perhaps both are acceptable.Alternatively, if we consider that the chef wants to minimize the area, he would choose the smaller ( B ), which is ( frac{alpha - sqrt{alpha^2 - 4 beta P}}{2 beta} ), leading to a smaller ( A ).But without explicit instructions, it's safer to provide both solutions.So, my final answers are:1. ( k = frac{n}{m} )2. ( B = frac{alpha pm sqrt{alpha^2 - 4 beta P}}{2 beta} ), and ( A = frac{m}{n} B )</think>

🧠:<think>Okay, so I have these two sub-problems to solve, both related to graph theory and combinatorics. Let me try to tackle them one by one.Starting with Sub-problem 1: Professor Jane wants the number of distinct topological sorts of a directed acyclic graph (DAG) with n nodes, where n can be up to 20. Hmm, topological sorting is something I remember from my algorithms class. A topological sort is an ordering of the nodes where for every directed edge from node u to node v, u comes before v in the ordering. And the number of such orderings can vary depending on the structure of the DAG.I think the straightforward way to compute the number of topological sorts is using a recursive approach. For each node with no incoming edges (a source node), we can choose it as the first node in the topological order and then recursively compute the number of topological sorts for the remaining graph. But wait, with n up to 20, a naive recursive approach might be too slow because the number of possibilities could be huge. For example, a linear chain of 20 nodes has only one topological sort, but a graph with all nodes disconnected would have 20! possible sorts, which is a massive number. So, we need a more efficient method.I recall that dynamic programming (DP) can be useful here. The idea is to memoize the number of topological sorts for subgraphs. Specifically, we can represent the state as a bitmask indicating which nodes have been included so far. For each state, we can iterate over all possible next nodes that can be added (i.e., nodes with all their predecessors already included). The number of topological sorts for the current state is the sum of the number of sorts for each possible next state.But wait, the state space for n=20 would be 2^20, which is about a million. That's manageable, right? Because 2^20 is 1,048,576, which is a number that modern computers can handle without too much trouble. So, the DP approach seems feasible.Let me outline the steps:1. For each node, determine its in-degree and its dependencies.2. Use a bitmask to represent the set of nodes that have been included in the topological order so far.3. For each state (bitmask), if it hasn't been computed yet, iterate over all nodes not yet included. For each such node, check if all its predecessors have been included (i.e., all nodes it depends on are in the current state). If so, recursively compute the number of topological sorts for the state with this node added.4. Memoize the results to avoid redundant computations.This should work efficiently for n up to 20.Now, moving on to Sub-problem 2: We need to find the number of bases of a matroid defined on the edges of a complete bipartite graph K_{m,n}, where m ≤ n ≤ 15. The bases correspond to the number of perfect matchings in K_{m,n}.Wait, perfect matchings in a bipartite graph. I remember that in a complete bipartite graph K_{m,n}, a perfect matching exists only if m = n. But in this case, the problem states m ≤ n. So, if m ≠ n, there are no perfect matchings. But if m = n, the number of perfect matchings is n! because each node on the left can be matched to any node on the right, and it's a bijection.But hold on, the matroid is defined on the edges of K_{m,n}. The bases of a matroid are the maximal independent sets. In the context of bipartite graphs, if we're considering the matroid where independent sets are matchings, then the bases would indeed be the perfect matchings when m = n. However, if m < n, the maximum size of a matching is m, so the bases would be all matchings of size m.Wait, but the problem says "the number of bases of this matroid, which corresponds to finding the number of perfect matchings in the bipartite graph K_{m,n}." Hmm, that wording is a bit confusing. If m ≤ n, then a perfect matching is only possible if m = n. Otherwise, the maximum matching size is m, which is less than n. So, if m ≠ n, there are no perfect matchings, so the number is zero. If m = n, then it's n!.But let me think again. Maybe the matroid is defined differently. In a bipartite graph, the edge set can form a matroid where the independent sets are the matchings. The rank function would be the size of the largest matching. So, the bases are the matchings of size equal to the rank, which is the size of the maximum matching. So, if m ≤ n, the maximum matching size is m, so the bases are all matchings of size m. But wait, that's not perfect matchings unless m = n.So, the number of bases is equal to the number of maximum matchings, which is the number of ways to choose m edges such that no two share a vertex. In K_{m,n}, the number of such maximum matchings is m! * C(n, m), but wait, no. Actually, for K_{m,n}, the number of perfect matchings when m = n is n!. If m < n, the number of maximum matchings is the number of ways to choose m edges with no two sharing a vertex. That would be the number of injective functions from the m nodes on one side to the n nodes on the other side, which is P(n, m) = n! / (n - m)!.Wait, let me verify. In K_{m,n}, the number of matchings of size k is C(m, k) * C(n, k) * k! because you choose k nodes from each partition and then match them in k! ways. So, for maximum matchings where k = m (since m ≤ n), it would be C(m, m) * C(n, m) * m! = 1 * C(n, m) * m! = P(n, m) = n! / (n - m)!.But in the problem statement, it says "the number of bases of this matroid, which corresponds to finding the number of perfect matchings in the bipartite graph K_{m,n}." So, maybe the problem is assuming that m = n? Or perhaps it's a misstatement, and they actually mean maximum matchings.Wait, let's read it again: "the number of bases of this matroid, which corresponds to finding the number of perfect matchings in the bipartite graph K_{m,n}." So, the correspondence is to perfect matchings. Therefore, perhaps the matroid is such that the bases are the perfect matchings. But in K_{m,n}, perfect matchings only exist when m = n. So, if m ≠ n, the number is zero. If m = n, it's n!.But the problem says m ≤ n ≤ 15. So, if m < n, the number of perfect matchings is zero. If m = n, it's n!.Alternatively, maybe the matroid is defined differently. For example, in a graphic matroid, the independent sets are the forests, but here it's a bipartite graph. Maybe it's a matching matroid, where the independent sets are the matchings. In that case, the bases are the maximum matchings, which have size min(m, n). So, if m ≤ n, the maximum matching size is m, and the number of such bases is the number of maximum matchings, which is P(n, m) = n! / (n - m)!.But the problem explicitly says "perfect matchings," which implies that m = n. So, perhaps the problem is only considering the case when m = n, but the constraints say m ≤ n. Hmm, this is a bit confusing.Wait, maybe the problem is using "perfect matching" in a different sense. In bipartite graphs, a perfect matching is a matching that covers all vertices, which requires that m = n. So, if m ≠ n, there are no perfect matchings. Therefore, the number of perfect matchings is zero when m ≠ n, and n! when m = n.But the problem says "the number of bases of this matroid, which corresponds to finding the number of perfect matchings in the bipartite graph K_{m,n}." So, perhaps the matroid is such that the bases are the perfect matchings, which only exist when m = n. So, the answer is n! if m = n, else zero.But the problem doesn't specify that m = n, just that m ≤ n. So, perhaps the answer is zero unless m = n, in which case it's n!.Alternatively, maybe the matroid is defined such that the bases are the maximum matchings, regardless of whether they are perfect or not. In that case, the number of bases is P(n, m) = n! / (n - m)!.But the problem explicitly says "perfect matchings," so I think it's safer to assume that m = n. Therefore, the number of perfect matchings is n!.Wait, but the problem says "the number of bases of this matroid, which corresponds to finding the number of perfect matchings in the bipartite graph K_{m,n}." So, perhaps the matroid is such that the bases are the perfect matchings, which only exist when m = n. So, if m ≠ n, the number is zero. If m = n, it's n!.But the problem doesn't specify that m = n, just that m ≤ n. So, perhaps the answer is zero unless m = n, in which case it's n!.Alternatively, maybe the problem is considering that a perfect matching in K_{m,n} is a matching that covers all nodes on the smaller side, which is m. So, in that case, the number of perfect matchings is P(n, m) = n! / (n - m)!.Wait, I'm getting confused. Let me clarify:In a bipartite graph K_{m,n}, a perfect matching is a set of edges such that every node is matched. This is only possible if m = n. If m ≠ n, there are no perfect matchings.However, sometimes people refer to a "perfect matching" in a bipartite graph as a matching that covers all nodes on one side, especially when the sides are of unequal size. In that case, if m ≤ n, a perfect matching would be a matching that covers all m nodes on the smaller side, leaving n - m nodes on the larger side unmatched. In that case, the number of such perfect matchings would be P(n, m) = n! / (n - m)!.But the standard definition requires that all nodes are matched, which only happens when m = n. So, the problem might be using the standard definition, in which case the number is zero unless m = n, else n!.But the problem says "the number of bases of this matroid, which corresponds to finding the number of perfect matchings in the bipartite graph K_{m,n}." So, perhaps the matroid is such that the bases are the perfect matchings, which only exist when m = n. Therefore, the number is n! if m = n, else zero.Alternatively, if the matroid is defined such that the bases are the maximum matchings, which have size min(m, n), then the number is P(n, m) = n! / (n - m)!.I think I need to clarify this. Let me recall that in a bipartite graph, the edge set forms a matroid where the independent sets are the matchings. The rank of the matroid is the size of the maximum matching, which is min(m, n). Therefore, the bases are the matchings of size min(m, n). So, if m ≤ n, the bases are the matchings of size m, which are the maximum matchings. The number of such bases is the number of ways to choose m edges such that no two share a vertex, which is P(n, m) = n! / (n - m)!.But the problem says "the number of bases of this matroid, which corresponds to finding the number of perfect matchings in the bipartite graph K_{m,n}." So, perhaps the problem is conflating perfect matchings with maximum matchings. If that's the case, then the number is P(n, m).But I'm not entirely sure. Let me think of an example. If m = 2, n = 3. Then K_{2,3} has maximum matchings of size 2. The number of such matchings is P(3, 2) = 3 * 2 = 6. On the other hand, perfect matchings in K_{2,3} would require m = n, which isn't the case here, so there are zero perfect matchings.Therefore, the number of bases is 6 in this case, which is the number of maximum matchings, not perfect matchings. So, the problem might have a misstatement, and the correspondence is to maximum matchings, not perfect matchings.Alternatively, if the problem is considering perfect matchings as those that cover all nodes on the smaller side, then it's the same as maximum matchings. So, perhaps the problem is using "perfect matching" in that sense.Given that, the number of bases is P(n, m) = n! / (n - m)!.But let me check another example. If m = 3, n = 3, then P(n, m) = 6, which is 3!, which is the number of perfect matchings. So, in that case, it aligns with the standard definition.Wait, so when m = n, P(n, m) = n! which is the number of perfect matchings. When m < n, P(n, m) is the number of maximum matchings, which are sometimes called "perfect" in the sense that they cover all nodes on the smaller side.Therefore, perhaps the problem is using "perfect matching" in the sense of maximum matching, regardless of whether both sides are covered. So, the number of bases is P(n, m) = n! / (n - m)!.But I'm not entirely certain. The problem statement says "perfect matchings," which traditionally require both partitions to be covered, hence m = n. So, perhaps the answer is zero unless m = n, in which case it's n!.But given that the problem mentions m ≤ n, and doesn't specify m = n, I think it's safer to assume that the number of bases is the number of maximum matchings, which is P(n, m) = n! / (n - m)!.Wait, but in the problem statement, it says "the number of bases of this matroid, which corresponds to finding the number of perfect matchings in the bipartite graph K_{m,n}." So, if the correspondence is to perfect matchings, then the number is zero unless m = n, else n!.But I'm still confused because the problem allows m ≤ n, not necessarily m = n.Alternatively, perhaps the matroid is such that the bases are the perfect matchings, but in K_{m,n}, perfect matchings only exist when m = n, so the number is n! if m = n, else zero.Given that, I think the answer is:If m = n, the number of perfect matchings is n!.If m ≠ n, the number is zero.But the problem says m ≤ n, so the answer is zero unless m = n, in which case it's n!.Alternatively, if the problem is considering maximum matchings as perfect matchings, then the number is P(n, m).I think I need to make a decision here. Given that the problem explicitly mentions "perfect matchings," which traditionally require m = n, I will go with that interpretation. Therefore, the number of bases is n! if m = n, else zero.But wait, the problem says "the number of bases of this matroid, which corresponds to finding the number of perfect matchings in the bipartite graph K_{m,n}." So, perhaps the matroid is such that the bases are the perfect matchings, which only exist when m = n. Therefore, the number is n! if m = n, else zero.Alternatively, if the matroid is defined such that the bases are the maximum matchings, which are sometimes called perfect matchings in the context of bipartite graphs when they cover all nodes on one side, then the number is P(n, m).I think I need to go with the standard definition. Therefore, the number of perfect matchings is n! if m = n, else zero.But wait, let me think again. In the context of matroids, the bases are the maximal independent sets. In the matching matroid, the independent sets are the matchings, and the bases are the maximum matchings. So, the number of bases is the number of maximum matchings, which is P(n, m) when m ≤ n.Therefore, the number of bases is P(n, m) = n! / (n - m)!.But the problem says "the number of bases of this matroid, which corresponds to finding the number of perfect matchings in the bipartite graph K_{m,n}." So, perhaps the problem is using "perfect matchings" to mean maximum matchings, which would be P(n, m).Given that, I think the answer is P(n, m) = n! / (n - m)!.But I'm still not entirely sure. Let me check with m = 2, n = 3. The number of maximum matchings is 6, which is P(3, 2) = 6. The number of perfect matchings (in the traditional sense) is zero because m ≠ n. So, if the problem is referring to maximum matchings, the answer is 6. If it's referring to perfect matchings, the answer is zero.Given that the problem says "the number of bases of this matroid, which corresponds to finding the number of perfect matchings," I think it's more likely that they are referring to maximum matchings, because otherwise, the number would often be zero, which might be trivial.Therefore, I think the answer is P(n, m) = n! / (n - m)!.But to be safe, I'll note both interpretations.In summary:Sub-problem 1: Use dynamic programming with bitmasking to compute the number of topological sorts. The approach is to memoize the number of sorts for each subset of nodes, and for each subset, add the number of sorts for each possible next node that can be added.Sub-problem 2: The number of bases is the number of maximum matchings in K_{m,n}, which is P(n, m) = n! / (n - m)!.But wait, in the problem statement, it's mentioned that the matroid is defined on the edges of K_{m,n}. So, the ground set E is all edges, and the independent sets are the matchings. Therefore, the bases are the maximum matchings, which have size min(m, n). So, the number of bases is the number of maximum matchings, which is P(n, m) when m ≤ n.Yes, that makes sense. Therefore, the answer is P(n, m) = n! / (n - m)!.So, putting it all together:For Sub-problem 1, the number of topological sorts can be computed using dynamic programming with bitmasking, resulting in an efficient solution even for n=20.For Sub-problem 2, the number of bases is the number of maximum matchings in K_{m,n}, which is P(n, m) = n! / (n - m)!.But wait, let me double-check the formula. The number of ways to choose m edges in K_{m,n} such that no two share a vertex is indeed P(n, m). Because for each of the m nodes on one side, we can assign a unique node on the other side. The first node has n choices, the second has n-1, and so on, down to n - m + 1. So, it's n * (n - 1) * ... * (n - m + 1) = n! / (n - m)!.Yes, that's correct.So, final answers:Sub-problem 1: Use DP with bitmasking to compute the number of topological sorts.Sub-problem 2: The number of bases is P(n, m) = n! / (n - m)!.But wait, in the problem statement, it's K_{m,n}, so m is the number of nodes on one side, n on the other. So, if m ≤ n, the number of maximum matchings is P(n, m). If m > n, it would be P(m, n), but since m ≤ n, it's P(n, m).Yes, that's correct.So, to summarize:Sub-problem 1: The number of topological sorts can be computed efficiently using dynamic programming with bitmasking, resulting in a solution that handles n up to 20.Sub-problem 2: The number of bases is P(n, m) = n! / (n - m)!.</think>

🧠:<think>Alright, so I've got this problem about cryptography, and I need to solve two sub-problems related to RSA encryption and a one-time pad. Let me take it step by step.Starting with Sub-problem 1. The officer is using RSA, and they've chosen two prime numbers, p = 61 and q = 53. I need to calculate the modulus n and Euler's totient function φ(n). Then, given that the public exponent e is 17, I have to find the private key d such that d * e ≡ 1 mod φ(n). Okay, modulus n in RSA is just the product of p and q. So, n = p * q. Let me compute that. 61 multiplied by 53. Hmm, 60*50 is 3000, 60*3 is 180, 1*50 is 50, and 1*3 is 3. So adding those up: 3000 + 180 + 50 + 3 = 3233. So n is 3233.Next, Euler's totient function φ(n). Since n is the product of two distinct primes, φ(n) = (p - 1)*(q - 1). So, p is 61, so p - 1 is 60, and q is 53, so q - 1 is 52. Multiplying those together: 60 * 52. Let me do that. 60*50 is 3000, and 60*2 is 120, so 3000 + 120 = 3120. So φ(n) is 3120.Now, moving on to finding the private key d. We have e = 17, and we need d such that (d * e) mod φ(n) = 1. In other words, we need to find the modular inverse of e modulo φ(n). So, d ≡ e^{-1} mod φ(n). To find d, I can use the Extended Euclidean Algorithm, which finds integers x and y such that a*x + b*y = gcd(a, b). In this case, a is e (17) and b is φ(n) (3120). Since 17 and 3120 should be coprime (as e is chosen to be coprime with φ(n) in RSA), their gcd should be 1, and thus x will be the modular inverse.Let me set up the algorithm. I need to find x such that 17x ≡ 1 mod 3120.So, let's perform the Extended Euclidean Algorithm on 3120 and 17.First, divide 3120 by 17:3120 ÷ 17. Let's see, 17*183 = 3111, because 17*180 = 3060, and 17*3=51, so 3060 + 51 = 3111. Then, 3120 - 3111 = 9. So, 3120 = 17*183 + 9.Now, take 17 and divide by the remainder 9:17 ÷ 9 = 1 with a remainder of 8, since 9*1=9, and 17 - 9 = 8.So, 17 = 9*1 + 8.Next, divide 9 by 8:9 ÷ 8 = 1 with a remainder of 1, because 8*1=8, and 9 - 8 = 1.So, 9 = 8*1 + 1.Now, divide 8 by 1:8 ÷ 1 = 8 with a remainder of 0. So, we stop here since the remainder is 0.Now, working backwards to express 1 as a linear combination of 17 and 3120.From the last non-zero remainder, which is 1:1 = 9 - 8*1But 8 is from the previous step: 8 = 17 - 9*1Substitute that into the equation:1 = 9 - (17 - 9*1)*1 = 9 - 17 + 9 = 2*9 - 17But 9 is from the first step: 9 = 3120 - 17*183Substitute that in:1 = 2*(3120 - 17*183) - 17 = 2*3120 - 2*17*183 - 17Simplify:1 = 2*3120 - (2*183 + 1)*17Calculate 2*183 + 1: 366 + 1 = 367So, 1 = 2*3120 - 367*17This means that -367*17 ≡ 1 mod 3120Therefore, the coefficient of 17 is -367, which is the modular inverse. But we need a positive value for d, so we add 3120 to -367 until we get a positive number.Compute -367 + 3120 = 2753So, d = 2753.Let me verify that 17 * 2753 mod 3120 is indeed 1.17 * 2753: Let me compute 2753 * 17.2753 * 10 = 275302753 * 7 = 19271Add them together: 27530 + 19271 = 46801Now, divide 46801 by 3120 to find the remainder.3120 * 15 = 46800So, 46801 - 46800 = 1Thus, 46801 mod 3120 is 1. So yes, 17 * 2753 ≡ 1 mod 3120. That checks out.So, for Sub-problem 1, n is 3233, φ(n) is 3120, and d is 2753.Moving on to Sub-problem 2. They're using a one-time pad for additional encryption. The message M is a binary string of length 8: 11010101. The ciphertext C is 01101111. I need to find the key K.In a one-time pad, the ciphertext is obtained by XORing the message with the key. So, C = M XOR K. To find K, we can rearrange this: K = M XOR C.So, let's perform the XOR operation bit by bit.M: 1 1 0 1 0 1 0 1C: 0 1 1 0 1 1 1 1Let me write them down:M: 1 1 0 1 0 1 0 1C: 0 1 1 0 1 1 1 1XOR each corresponding bit:1 XOR 0 = 11 XOR 1 = 00 XOR 1 = 11 XOR 0 = 10 XOR 1 = 11 XOR 1 = 00 XOR 1 = 11 XOR 1 = 0So, putting it all together: 1 0 1 1 1 0 1 0So, K is 10111010.Let me double-check that. If I XOR M and K, do I get C?M: 11010101K:10111010XOR:1^1=01^0=10^1=11^1=00^1=11^0=10^1=11^0=1Wait, that gives 01101111, which is C. Perfect, so K is indeed 10111010.Now, the second part of Sub-problem 2 asks why the one-time pad is considered unbreakable if the key is truly random and used only once.Well, the one-time pad is unbreakable because it's provably secure. The key is as long as the message, and it's used only once. Since the key is random, each bit of the ciphertext gives no information about the corresponding bit of the message. Without the key, an attacker cannot determine the message because every possible message of the same length is a possible decryption. There's no statistical pattern to exploit, and brute-force would require trying all possible keys, which is infeasible. So, as long as the key is truly random, kept secret, and not reused, the one-time pad provides perfect secrecy.Let me just make sure I didn't make any mistakes in the calculations. For Sub-problem 1, n = 61*53=3233, φ(n)=60*52=3120. Then, using the Extended Euclidean Algorithm, I found d=2753. That seems correct.For Sub-problem 2, XORing M and C gives K=10111010. That also seems correct. And the explanation about the one-time pad's security is solid.I think that's all. I don't see any errors in my reasoning.Final AnswerSub-problem 1:1. ( n = boxed{3233} ) and ( phi(n) = boxed{3120} ).2. The private key ( d = boxed{2753} ).Sub-problem 2:1. The one-time pad key ( K = boxed{10111010} ).2. The one-time pad is unbreakable because it provides perfect secrecy when the key is truly random and used only once.</think>

🧠:<think>Alright, so I have this problem about a library manager who curates a book club. There are two parts to the problem. Let me try to tackle them one by one.Starting with part 1. The manager wants to maintain a ratio of Fiction to Non-Fiction to Mystery books as 3:2:1 each month. The total number of books selected is T. I need to find expressions for F, N, and M in terms of T.Hmm, ratios can sometimes be tricky, but I remember that ratios can be converted into fractions of the total. So, if the ratio is 3:2:1, that means for every 3 Fiction books, there are 2 Non-Fiction and 1 Mystery. The total parts of the ratio would be 3 + 2 + 1, which is 6 parts.So, each part is equal to T divided by 6. Therefore, Fiction would be 3 parts, which is (3/6)T, Non-Fiction is 2 parts, which is (2/6)T, and Mystery is 1 part, which is (1/6)T.Wait, let me write that down more clearly:Total ratio parts = 3 + 2 + 1 = 6.So,F = (3/6) * T = (1/2)T,N = (2/6) * T = (1/3)T,M = (1/6) * T.Let me check if that makes sense. If T is 6, then F would be 3, N would be 2, and M would be 1, which fits the ratio. If T is 12, then F is 6, N is 4, M is 2. Yep, that seems right.So, part 1 is done. Now, moving on to part 2.The manager introduces a new book every month that increases the total number of books by a factor of k, where k is 1.1. The initial total number of books is T0 = 24. We need to calculate the total number of books selected by the end of 12 months.Wait, does this mean that each month, the total number of books is multiplied by 1.1? So, it's a geometric sequence where each term is 1.1 times the previous term.So, the total number of books each month would be:Month 1: T1 = T0 * k = 24 * 1.1,Month 2: T2 = T1 * k = 24 * (1.1)^2,...Month 12: T12 = 24 * (1.1)^12.But the question is asking for the total number of books selected by the end of 12 months. So, that would be the sum of T0, T1, T2, ..., T12? Wait, hold on.Wait, actually, the initial total is T0 = 24. Then each subsequent month, the total increases by a factor of k. So, the total number of books each month is:Month 1: T1 = T0 * k,Month 2: T2 = T1 * k = T0 * k^2,...Month 12: T12 = T0 * k^12.But the question is asking for the total number of books selected by the end of 12 months. So, that would be the sum of all the monthly totals from month 1 to month 12.Wait, but hold on, is T0 included? The initial total is T0 = 24, which is presumably the first month. Then, each subsequent month, it's multiplied by k. So, month 1: T1 = 24,month 2: T2 = 24 * 1.1,month 3: T3 = 24 * (1.1)^2,...month 12: T12 = 24 * (1.1)^11.Wait, now I'm confused. Is the initial total T0 the first month, and then each next month is multiplied by k? So, month 1: T0 = 24,month 2: T1 = 24 * 1.1,month 3: T2 = 24 * (1.1)^2,...month 12: T11 = 24 * (1.1)^11.Therefore, the total number of books over 12 months would be the sum from n=0 to n=11 of T0 * k^n.Which is a geometric series with first term a = 24, common ratio r = 1.1, and number of terms N = 12.The formula for the sum of a geometric series is S = a*(r^N - 1)/(r - 1).So, plugging in the numbers:S = 24*(1.1^12 - 1)/(1.1 - 1).Let me compute that.First, compute 1.1^12. Let me see, 1.1^12 is approximately... Hmm, 1.1^1 is 1.1,1.1^2 = 1.21,1.1^3 = 1.331,1.1^4 = 1.4641,1.1^5 = 1.61051,1.1^6 = 1.771561,1.1^7 = 1.9487171,1.1^8 = 2.14358881,1.1^9 = 2.357947691,1.1^10 = 2.5937424601,1.1^11 = 2.85311670611,1.1^12 = 3.138428376721.So, approximately 3.138428376721.Therefore, S = 24*(3.138428376721 - 1)/(0.1).Compute numerator: 3.138428376721 - 1 = 2.138428376721.Multiply by 24: 24 * 2.138428376721 ≈ 24 * 2.138428 ≈ let's compute 24*2 = 48, 24*0.138428 ≈ 24*0.138 ≈ 3.312, so total ≈ 48 + 3.312 ≈ 51.312.Then divide by 0.1: 51.312 / 0.1 = 513.12.So, approximately 513.12 books. But since we can't have a fraction of a book, we might need to round it. However, the problem doesn't specify whether to round or not, so perhaps we can leave it as is.Wait, but let me double-check my calculations because 1.1^12 is approximately 3.138428, so 3.138428 - 1 = 2.138428.2.138428 * 24 = Let me compute 2 * 24 = 48, 0.138428 * 24 ≈ 3.322272. So, total ≈ 48 + 3.322272 ≈ 51.322272.Then, 51.322272 / 0.1 = 513.22272.So, approximately 513.22 books. So, about 513 books.But let me verify if I used the correct number of terms. The initial total is T0 = 24, which is month 1. Then, each subsequent month up to month 12. So, that's 12 terms. So, n=0 to n=11, which is 12 terms.So, the formula is correct.Alternatively, maybe the problem is interpreted differently. Maybe the total number of books each month is multiplied by k, starting from T0. So, the first month is T0, the second month is T0*k, third is T0*k^2, ..., twelfth month is T0*k^11.So, the total is T0*(1 + k + k^2 + ... + k^11). Which is the same as the sum from n=0 to n=11 of T0*k^n.Which is what I computed earlier.So, 24*(1.1^12 - 1)/(1.1 - 1) ≈ 24*(3.138428 - 1)/0.1 ≈ 24*2.138428/0.1 ≈ 24*21.38428 ≈ 513.22272.So, approximately 513.22 books. Since we can't have a fraction, maybe we round to the nearest whole number, which is 513.Alternatively, if we keep more decimal places, perhaps it's 513.22, but in the context of books, it's likely to be rounded to 513.Wait, but let me check my calculation of 1.1^12 again because that's crucial.1.1^1 = 1.11.1^2 = 1.211.1^3 = 1.3311.1^4 = 1.46411.1^5 = 1.610511.1^6 = 1.7715611.1^7 = 1.94871711.1^8 = 2.143588811.1^9 = 2.3579476911.1^10 = 2.59374246011.1^11 = 2.853116706111.1^12 = 3.138428376721Yes, that seems correct.So, 3.138428376721 - 1 = 2.1384283767212.138428376721 * 24 = Let me compute this more accurately.2.138428376721 * 24:First, 2 * 24 = 480.138428376721 * 24:0.1 * 24 = 2.40.038428376721 * 24 ≈ 0.038428376721 * 24 ≈ 0.9222810413064So, total ≈ 2.4 + 0.922281 ≈ 3.322281So, total ≈ 48 + 3.322281 ≈ 51.322281Then, 51.322281 / 0.1 = 513.22281So, approximately 513.22281, which is about 513.22.So, 513.22 books. Since we can't have a fraction, we might round to 513 books.Alternatively, if we consider that each month's total must be an integer, but the problem doesn't specify that, so perhaps we can leave it as 513.22, but in the context of the problem, it's more likely to be rounded to the nearest whole number.Therefore, the total number of books selected by the end of 12 months is approximately 513.Wait, but let me make sure that the initial interpretation is correct. The problem says: "the manager decides to introduce a new book every month that increases the total number of books selected each month by a factor of k, such that T_next = k * T_current."So, starting from T0 = 24, each subsequent month's total is k times the previous month's total. So, month 1: T1 = 24 * 1.1,month 2: T2 = 24 * (1.1)^2,...month 12: T12 = 24 * (1.1)^11.Wait, hold on, if we start counting from month 1 as T0, then month 12 would be T11. So, the total number of books over 12 months would be the sum from n=0 to n=11 of 24*(1.1)^n.Which is what I calculated earlier, giving approximately 513.22.Alternatively, if the initial total T0 is month 0, and we're calculating up to month 12, that would be 13 terms, but the problem says "by the end of 12 months," so likely 12 terms.Wait, let me read the problem again:"For the book club's annual plan, the manager decides to introduce a new book every month that increases the total number of books selected each month by a factor of k, such that T_next = k * T_current. Assuming the initial total number of books T0 is 24 and k = 1.1, calculate the total number of books selected by the end of 12 months."So, starting from T0 = 24, which is presumably the first month, then each next month is multiplied by k. So, month 1: T1 = 24*1.1,month 2: T2 = 24*(1.1)^2,...month 12: T12 = 24*(1.1)^11.So, the total is the sum from n=0 to n=11 of 24*(1.1)^n.Which is 24*(1.1^12 - 1)/(1.1 - 1) ≈ 513.22.So, I think that's correct.Therefore, the answer is approximately 513 books.But let me check if I can compute it more precisely.Using the formula S = a*(r^n - 1)/(r - 1),where a = 24,r = 1.1,n = 12.So, S = 24*(1.1^12 - 1)/(0.1).Compute 1.1^12:As above, 1.1^12 ≈ 3.138428376721.So, 3.138428376721 - 1 = 2.138428376721.2.138428376721 / 0.1 = 21.38428376721.21.38428376721 * 24 = ?Let me compute 20 * 24 = 480,1.38428376721 * 24 ≈ let's compute 1 * 24 = 24,0.38428376721 * 24 ≈ 9.222810413064.So, total ≈ 24 + 9.222810413064 ≈ 33.222810413064.So, total S ≈ 480 + 33.222810413064 ≈ 513.222810413064.So, approximately 513.2228 books.So, rounding to the nearest whole number, 513 books.Alternatively, if we need to keep it as a decimal, 513.22, but since books are whole items, 513 is the appropriate answer.Therefore, the total number of books selected by the end of 12 months is approximately 513.So, summarizing:1. F = (1/2)T, N = (1/3)T, M = (1/6)T.2. Total books over 12 months ≈ 513.Final Answer1. The number of Fiction, Non-Fiction, and Mystery books are (boxed{frac{T}{2}}), (boxed{frac{T}{3}}), and (boxed{frac{T}{6}}) respectively.2. The total number of books selected by the end of 12 months is (boxed{513}).</think>