🧠:<think>Alright, so I've got this problem from Emily, and it's got two parts. Let me try to tackle them one by one. I'll start with the first part, which involves a sequence and encoding a word.First, Emily defines a sequence ( a_n ) where ( a_1 = 1 ) and for ( n geq 2 ), each term is the previous term plus ( n^2 ). So, the recursive formula is ( a_n = a_{n-1} + n^2 ). She then assigns each letter a number based on its position in the alphabet—A=1, B=2, ..., Z=26. Then, she encodes a word by summing the corresponding ( a_n ) values for each letter. For example, "BOOK" is encoded as ( a_2 + a_{15} + a_{15} + a_{11} ). My task is to find the encoded value for "INSPIRE".Okay, so first, I need to figure out the numerical values for each letter in "INSPIRE". Let's break it down:I - that's the 9th letter.N - the 14th.S - 19th.P - 16th.I - 9th again.R - 18th.E - 5th.So, the letters correspond to the numbers: 9, 14, 19, 16, 9, 18, 5.Now, I need to compute ( a_9 + a_{14} + a_{19} + a_{16} + a_9 + a_{18} + a_5 ). But to do that, I need to find each ( a_n ) term.Given that ( a_1 = 1 ), and each subsequent term is the previous term plus ( n^2 ). So, let's try to find a general formula for ( a_n ). Maybe it's a known sequence or can be expressed in terms of a summation.Since ( a_n = a_{n-1} + n^2 ), this is a recursive sequence where each term is built by adding the square of the term's index. So, expanding this, we can write:( a_n = a_1 + sum_{k=2}^{n} k^2 )But ( a_1 = 1 ), so:( a_n = 1 + sum_{k=2}^{n} k^2 )We know that the sum of squares from 1 to n is ( frac{n(n+1)(2n+1)}{6} ). So, if we subtract the first term (which is 1^2 = 1) from that sum, we get the sum from 2 to n.Therefore, ( sum_{k=2}^{n} k^2 = frac{n(n+1)(2n+1)}{6} - 1 )So, substituting back into the expression for ( a_n ):( a_n = 1 + left( frac{n(n+1)(2n+1)}{6} - 1 right) )Simplify that:( a_n = frac{n(n+1)(2n+1)}{6} )Wait, because the 1 and -1 cancel out. So, actually, ( a_n = frac{n(n+1)(2n+1)}{6} ). That's the formula for the sum of squares from 1 to n. Interesting, so ( a_n ) is just the sum of squares up to n. That makes sense because each term is adding the square of n.So, now, knowing that, I can compute each ( a_n ) term by plugging into this formula.Let me compute each term needed:First, let's compute ( a_5 ):( a_5 = frac{5 times 6 times 11}{6} = frac{330}{6} = 55 )Wait, let me check that:Wait, ( 5 times 6 = 30 ), ( 30 times 11 = 330 ), divided by 6 is 55. Yep, that's correct.Next, ( a_9 ):( a_9 = frac{9 times 10 times 19}{6} )Compute numerator: 9*10=90, 90*19=1710Divide by 6: 1710 / 6 = 285So, ( a_9 = 285 )Next, ( a_{14} ):( a_{14} = frac{14 times 15 times 29}{6} )Compute numerator: 14*15=210, 210*29=6090Divide by 6: 6090 / 6 = 1015So, ( a_{14} = 1015 )Next, ( a_{16} ):( a_{16} = frac{16 times 17 times 33}{6} )Compute numerator: 16*17=272, 272*33=8976Divide by 6: 8976 / 6 = 1496So, ( a_{16} = 1496 )Next, ( a_{18} ):( a_{18} = frac{18 times 19 times 37}{6} )Compute numerator: 18*19=342, 342*37=12654Divide by 6: 12654 / 6 = 2109So, ( a_{18} = 2109 )Next, ( a_{19} ):( a_{19} = frac{19 times 20 times 39}{6} )Compute numerator: 19*20=380, 380*39=14820Divide by 6: 14820 / 6 = 2470So, ( a_{19} = 2470 )Alright, so now I have all the necessary ( a_n ) terms:- ( a_5 = 55 )- ( a_9 = 285 )- ( a_{14} = 1015 )- ( a_{16} = 1496 )- ( a_{18} = 2109 )- ( a_{19} = 2470 )Wait, but in the word "INSPIRE", the letters correspond to 9,14,19,16,9,18,5. So, the sequence is:I (9), N (14), S (19), P (16), I (9), R (18), E (5)So, the encoded value is ( a_9 + a_{14} + a_{19} + a_{16} + a_9 + a_{18} + a_5 )Let me write that out:285 (I) + 1015 (N) + 2470 (S) + 1496 (P) + 285 (I) + 2109 (R) + 55 (E)Now, let's compute this step by step.First, add 285 + 1015:285 + 1015 = 1300Next, add 2470:1300 + 2470 = 3770Next, add 1496:3770 + 1496 = 5266Next, add 285:5266 + 285 = 5551Next, add 2109:5551 + 2109 = 7660Finally, add 55:7660 + 55 = 7715So, the encoded sequence value for "INSPIRE" is 7715.Wait, let me double-check the addition to make sure I didn't make a mistake.Start with 285 + 1015 = 1300. That's correct.1300 + 2470: 1300 + 2000 = 3300, 3300 + 470 = 3770. Correct.3770 + 1496: 3770 + 1000 = 4770, 4770 + 496 = 5266. Correct.5266 + 285: 5266 + 200 = 5466, 5466 + 85 = 5551. Correct.5551 + 2109: 5551 + 2000 = 7551, 7551 + 109 = 7660. Correct.7660 + 55: 7660 + 50 = 7710, 7710 + 5 = 7715. Correct.So, yes, 7715 is the total.Alright, so part 1 is done. Now, moving on to part 2.Emily creates a new sequence ( b_n ) where ( b_1 = 1 ), ( b_2 = 1 ), and for ( n geq 3 ), ( b_n = b_{n-1} + b_{n-2} + n ). She wants to find the first term ( b_k ) that is greater than the encoded value from part 1, which is 7715. So, I need to compute terms of this sequence until I find the smallest ( k ) such that ( b_k > 7715 ).First, let's understand the sequence ( b_n ). It's similar to the Fibonacci sequence but with an additional term ( n ) added each time. So, each term is the sum of the two previous terms plus the current index.Given that, let's try to compute the terms step by step until we exceed 7715.Given:- ( b_1 = 1 )- ( b_2 = 1 )For ( n geq 3 ):( b_n = b_{n-1} + b_{n-2} + n )Let me compute the terms one by one.Compute ( b_3 ):( b_3 = b_2 + b_1 + 3 = 1 + 1 + 3 = 5 )( b_4 = b_3 + b_2 + 4 = 5 + 1 + 4 = 10 )( b_5 = b_4 + b_3 + 5 = 10 + 5 + 5 = 20 )( b_6 = b_5 + b_4 + 6 = 20 + 10 + 6 = 36 )( b_7 = b_6 + b_5 + 7 = 36 + 20 + 7 = 63 )( b_8 = b_7 + b_6 + 8 = 63 + 36 + 8 = 107 )( b_9 = b_8 + b_7 + 9 = 107 + 63 + 9 = 179 )( b_{10} = b_9 + b_8 + 10 = 179 + 107 + 10 = 296 )( b_{11} = b_{10} + b_9 + 11 = 296 + 179 + 11 = 486 )( b_{12} = b_{11} + b_{10} + 12 = 486 + 296 + 12 = 794 )( b_{13} = b_{12} + b_{11} + 13 = 794 + 486 + 13 = 1293 )( b_{14} = b_{13} + b_{12} + 14 = 1293 + 794 + 14 = 2101 )( b_{15} = b_{14} + b_{13} + 15 = 2101 + 1293 + 15 = 3409 )( b_{16} = b_{15} + b_{14} + 16 = 3409 + 2101 + 16 = 5526 )( b_{17} = b_{16} + b_{15} + 17 = 5526 + 3409 + 17 = 8952 )Wait, hold on. ( b_{17} = 5526 + 3409 + 17 ). Let me compute that:5526 + 3409: 5526 + 3000 = 8526, 8526 + 409 = 89358935 + 17 = 8952So, ( b_{17} = 8952 )But we need ( b_k > 7715 ). So, let's check ( b_{16} = 5526 ), which is less than 7715. Then ( b_{17} = 8952 ), which is greater than 7715.Therefore, the smallest ( k ) such that ( b_k > 7715 ) is 17.Wait, but let me double-check my calculations because sometimes when adding large numbers, it's easy to make a mistake.Let me recompute ( b_{15} ):( b_{15} = b_{14} + b_{13} + 15 = 2101 + 1293 + 15 )2101 + 1293: 2100 + 1293 = 3393, plus 1 is 33943394 + 15 = 3409. Correct.( b_{16} = b_{15} + b_{14} + 16 = 3409 + 2101 + 16 )3409 + 2101: 3400 + 2100 = 5500, 9 + 1 = 10, so 5500 + 10 = 55105510 + 16 = 5526. Correct.( b_{17} = b_{16} + b_{15} + 17 = 5526 + 3409 + 17 )5526 + 3409: Let's compute 5526 + 3000 = 8526, 8526 + 409 = 89358935 + 17 = 8952. Correct.So, yes, ( b_{17} = 8952 ), which is greater than 7715, and ( b_{16} = 5526 ), which is less than 7715. Therefore, the smallest ( k ) is 17.Wait, but just to make sure, let me compute ( b_{17} ) again:5526 (b16) + 3409 (b15) + 17 = 5526 + 3409 = 8935; 8935 +17=8952. Correct.So, 8952 is greater than 7715, so k=17 is the answer.But just to be thorough, let me compute ( b_{18} ) as well to see how the sequence progresses:( b_{18} = b_{17} + b_{16} + 18 = 8952 + 5526 + 18 )8952 + 5526: Let's compute 8000 + 5000 = 13000, 952 + 526 = 1478, so total is 13000 + 1478 = 1447814478 + 18 = 14496So, ( b_{18} = 14496 ), which is way larger, but since we already found that ( b_{17} ) is the first term exceeding 7715, we don't need to go further.Wait, but hold on, let me check if I computed ( b_{14} ) correctly because sometimes when I compute multiple terms, I might have messed up an earlier term.Let me go back step by step:( b_1 = 1 )( b_2 = 1 )( b_3 = 1 + 1 + 3 = 5 )( b_4 = 5 + 1 + 4 = 10 )( b_5 = 10 + 5 + 5 = 20 )( b_6 = 20 + 10 + 6 = 36 )( b_7 = 36 + 20 + 7 = 63 )( b_8 = 63 + 36 + 8 = 107 )( b_9 = 107 + 63 + 9 = 179 )( b_{10} = 179 + 107 + 10 = 296 )( b_{11} = 296 + 179 + 11 = 486 )( b_{12} = 486 + 296 + 12 = 794 )( b_{13} = 794 + 486 + 13 = 1293 )( b_{14} = 1293 + 794 + 14 = 2101 )( b_{15} = 2101 + 1293 + 15 = 3409 )( b_{16} = 3409 + 2101 + 16 = 5526 )( b_{17} = 5526 + 3409 + 17 = 8952 )All these steps seem correct. Each term is the sum of the two previous terms plus the current index. So, yes, ( b_{17} = 8952 ) is indeed the first term exceeding 7715.Therefore, the smallest ( k ) is 17.Final AnswerThe smallest value of ( k ) is boxed{17}.</think>

🧠:<think>Alright, so I've got this problem about a restaurant owner who wants to create a special "Wine Pairing Night" event. They have 30 wines on the menu: 18 organic and 12 biodynamic. The goal is to figure out how many different sequences of 5 wines they can create under certain constraints.Starting with the first question: They want each sequence to have at least 2 organic wines and at least 2 biodynamic wines. So, I need to calculate the number of possible sequences of 5 wines where there are at least 2 of each type.Hmm, okay. So, the total number of wines is 30, with 18 organic and 12 biodynamic. Since the order matters here because it's a sequence, I think this is a permutation problem. But since we're selecting from two different types with constraints, maybe I should break it down into cases.First, let's consider the possible distributions of organic and biodynamic wines in the sequence. Since each sequence must have at least 2 of each, the possible splits are:- 2 organic and 3 biodynamic- 3 organic and 2 biodynamicWait, is that all? Let me check. If we have 5 wines, and we need at least 2 of each, then the minimum for each is 2, so the remaining 1 can be either. So, yes, the two cases are 2-3 and 3-2.So, for each case, I can calculate the number of sequences and then add them together.Starting with the first case: 2 organic and 3 biodynamic.The number of ways to choose 2 organic wines out of 18 is C(18,2), and the number of ways to choose 3 biodynamic wines out of 12 is C(12,3). Then, since the order matters, we need to multiply by the number of ways to arrange these 5 wines. Since there are 2 organic and 3 biodynamic, the number of arrangements is 5! / (2!3!) because of the permutations of identical items.Wait, hold on. Actually, since each wine is distinct, maybe I don't need to divide by the factorials? Hmm, no, actually, no. Because each wine is unique, the number of arrangements is simply 5! regardless. Wait, no, that's not quite right.Wait, no, the formula for permutations of multiset is n! / (n1!n2!...nk!), where n is the total number, and n1, n2,...nk are the numbers of each type. So, in this case, n = 5, n1 = 2, n2 = 3. So, the number of distinct sequences is 5! / (2!3!) = 10.But wait, but each organic wine is unique, and each biodynamic wine is unique. So, actually, maybe I should think of it differently.First, choose the 2 organic wines: C(18,2). Then, choose the 3 biodynamic wines: C(12,3). Then, for each such combination, arrange them in a sequence. Since each wine is unique, the number of arrangements is 5!.Wait, but that would be C(18,2) * C(12,3) * 5!.But wait, no, that would be overcounting because once you choose the specific wines, the number of ways to arrange them is 5!.But actually, no, that's correct because each selection of 2 organic and 3 biodynamic wines can be arranged in 5! ways. So, the total number of sequences for this case is C(18,2) * C(12,3) * 5!.Similarly, for the case of 3 organic and 2 biodynamic, it's C(18,3) * C(12,2) * 5!.So, the total number of sequences is [C(18,2)*C(12,3) + C(18,3)*C(12,2)] * 5!.Wait, but hold on. Let me verify this approach.Alternatively, another way to think about it is: the total number of sequences without any constraints is P(30,5) = 30*29*28*27*26. Then, subtract the sequences that don't meet the criteria, i.e., sequences with fewer than 2 organic or fewer than 2 biodynamic.But that might be more complicated because we have to use inclusion-exclusion. Maybe the case method is better.Wait, but let me see. The total number of sequences is 30P5 = 30*29*28*27*26.Now, the number of sequences with fewer than 2 organic wines: that would be sequences with 0 or 1 organic wines.Similarly, the number of sequences with fewer than 2 biodynamic wines: 0 or 1.But since we need to subtract both, but we have to be careful not to subtract too much.But maybe the case method is simpler.So, going back, for the first case: 2 organic and 3 biodynamic.Number of ways: C(18,2) * C(12,3) * 5!.Similarly, for 3 organic and 2 biodynamic: C(18,3) * C(12,2) * 5!.So, total sequences = [C(18,2)*C(12,3) + C(18,3)*C(12,2)] * 5!.Let me compute these values.First, compute C(18,2):C(18,2) = 18*17/2 = 153.C(12,3) = 12*11*10/6 = 220.So, C(18,2)*C(12,3) = 153 * 220. Let me compute that.153 * 200 = 30,600.153 * 20 = 3,060.So, total is 30,600 + 3,060 = 33,660.Next, C(18,3):C(18,3) = 18*17*16/6 = (18*17*16)/6.18/6 = 3, so 3*17*16 = 3*272 = 816.C(12,2) = 12*11/2 = 66.So, C(18,3)*C(12,2) = 816 * 66.Let me compute that.800*66 = 52,800.16*66 = 1,056.So, total is 52,800 + 1,056 = 53,856.So, adding both cases: 33,660 + 53,856 = 87,516.Then, multiply by 5! = 120.So, total sequences: 87,516 * 120.Let me compute that.First, 87,516 * 100 = 8,751,600.87,516 * 20 = 1,750,320.So, total is 8,751,600 + 1,750,320 = 10,501,920.Wait, that seems quite large. Let me check if I did the calculations correctly.Wait, 87,516 * 120.Alternatively, 87,516 * 12 * 10.87,516 * 12: 87,516 * 10 = 875,160; 87,516 * 2 = 175,032. So, total is 875,160 + 175,032 = 1,050,192. Then, times 10 is 10,501,920. Yeah, that's correct.But wait, let me think again. Is this the correct approach?Because when we choose 2 organic and 3 biodynamic, and then arrange them, is that the same as just permuting all 5? Or is there a different way?Wait, no, because each selection of 2 organic and 3 biodynamic is a unique set, and then arranging them in order. So, yes, multiplying by 5! is correct.Alternatively, another way to compute it is: For each case, the number of sequences is P(18,2) * P(12,3) * number of ways to interleave them.Wait, but that might complicate things. Alternatively, think of it as:First, choose the positions for the organic wines. There are C(5,2) ways to choose which 2 positions out of 5 will be organic. Then, for each of those positions, choose an organic wine: P(18,2). Similarly, for the remaining 3 positions, choose biodynamic wines: P(12,3).So, the total number of sequences for 2 organic and 3 biodynamic is C(5,2) * P(18,2) * P(12,3).Similarly, for 3 organic and 2 biodynamic, it's C(5,3) * P(18,3) * P(12,2).So, let's compute that.First, C(5,2) = 10.P(18,2) = 18*17 = 306.P(12,3) = 12*11*10 = 1,320.So, for 2 organic and 3 biodynamic: 10 * 306 * 1,320.Compute 10 * 306 = 3,060.3,060 * 1,320.Let me compute 3,000 * 1,320 = 3,960,000.60 * 1,320 = 79,200.So, total is 3,960,000 + 79,200 = 4,039,200.Similarly, for 3 organic and 2 biodynamic:C(5,3) = 10.P(18,3) = 18*17*16 = 4,896.P(12,2) = 12*11 = 132.So, 10 * 4,896 * 132.10 * 4,896 = 48,960.48,960 * 132.Compute 48,960 * 100 = 4,896,000.48,960 * 30 = 1,468,800.48,960 * 2 = 97,920.So, total is 4,896,000 + 1,468,800 = 6,364,800 + 97,920 = 6,462,720.Adding both cases: 4,039,200 + 6,462,720 = 10,501,920.So, same result as before. So, that seems correct.Therefore, the answer to the first question is 10,501,920.Now, moving on to the second question: If the owner decides that no two consecutive wines can be of the same type, how many valid sequences of 5 wines can be formed?So, now, in addition to having at least 2 organic and 2 biodynamic wines, we also have the constraint that no two consecutive wines are of the same type.Wait, but actually, the problem says "no two consecutive wines in the sequence can be of the same type." So, it's a different constraint, but does it also require at least 2 of each type? I think so, because the first part was under the constraints of at least 2 of each, but the second part is a new policy, so it's an additional constraint.Wait, actually, the problem says: "how many valid sequences of 5 wines can be formed under this new policy?" So, the new policy is only the no two consecutive same type. But does it still require at least 2 of each type? Or is it a separate constraint?Wait, the original problem was about creating sequences with at least 2 organic and 2 biodynamic. Then, the second question is about introducing a new policy that no two consecutive can be the same type. So, I think the new policy is in addition to the original constraints. So, the sequences must have at least 2 organic and 2 biodynamic, and also no two consecutive of the same type.But wait, let me check the exact wording:"how many valid sequences of 5 wines can be formed under this new policy?"So, the new policy is the only constraint? Or is it in addition to the previous constraints?Hmm, the wording is a bit ambiguous. But I think it's safer to assume that it's in addition to the previous constraints, because the first question was about the original constraints, and the second is introducing a new policy on top of that.But to be thorough, let's consider both interpretations.First interpretation: The new policy is the only constraint, i.e., sequences of 5 wines where no two consecutive are of the same type, regardless of the number of organic and biodynamic.Second interpretation: The new policy is added to the original constraints, so sequences must have at least 2 organic, 2 biodynamic, and no two consecutive of the same type.Given that the first question was about the original constraints, and the second is introducing a new policy, it's more likely that the new policy is an additional constraint. So, the sequences must satisfy both: at least 2 organic, 2 biodynamic, and no two consecutive same type.So, we need to calculate the number of sequences of 5 wines where:- At least 2 organic and at least 2 biodynamic.- No two consecutive wines are of the same type.So, how do we approach this?First, since no two consecutive can be the same type, the sequence must alternate between organic and biodynamic. But since we have 5 wines, which is odd, the sequence can start with either organic or biodynamic, and then alternate.But wait, but we also need at least 2 of each type. So, let's see:If the sequence starts with organic, the pattern would be O, B, O, B, O. So, 3 organic and 2 biodynamic.If it starts with biodynamic, the pattern would be B, O, B, O, B. So, 3 biodynamic and 2 organic.So, in both cases, we have either 3 organic and 2 biodynamic or 3 biodynamic and 2 organic.But wait, the original constraint was at least 2 of each, so both cases satisfy that.So, the total number of sequences is the sum of sequences starting with organic and sequences starting with biodynamic.So, let's compute both.First, sequences starting with organic:Pattern: O, B, O, B, O.Number of organic wines: 3.Number of biodynamic wines: 2.So, number of ways to choose 3 organic wines from 18: P(18,3).Number of ways to choose 2 biodynamic wines from 12: P(12,2).Similarly, since the pattern is fixed (O, B, O, B, O), the arrangement is determined by the selection.So, total sequences starting with organic: P(18,3) * P(12,2).Similarly, sequences starting with biodynamic:Pattern: B, O, B, O, B.Number of biodynamic wines: 3.Number of organic wines: 2.So, number of ways: P(12,3) * P(18,2).Therefore, total sequences: P(18,3)*P(12,2) + P(12,3)*P(18,2).Compute these values.First, P(18,3) = 18*17*16 = 4,896.P(12,2) = 12*11 = 132.So, P(18,3)*P(12,2) = 4,896 * 132.Compute that:4,896 * 100 = 489,600.4,896 * 30 = 146,880.4,896 * 2 = 9,792.So, total is 489,600 + 146,880 = 636,480 + 9,792 = 646,272.Next, P(12,3) = 12*11*10 = 1,320.P(18,2) = 18*17 = 306.So, P(12,3)*P(18,2) = 1,320 * 306.Compute that:1,320 * 300 = 396,000.1,320 * 6 = 7,920.So, total is 396,000 + 7,920 = 403,920.Adding both cases: 646,272 + 403,920 = 1,050,192.So, the total number of valid sequences is 1,050,192.Wait, let me verify that.Alternatively, another way to think about it is:For sequences starting with organic:- Choose 3 organic wines: P(18,3).- Choose 2 biodynamic wines: P(12,2).- Arrange them in the pattern O, B, O, B, O: which is fixed, so no additional permutations needed.Similarly, for sequences starting with biodynamic:- Choose 3 biodynamic wines: P(12,3).- Choose 2 organic wines: P(18,2).- Arrange them in the pattern B, O, B, O, B: fixed.So, yes, the total is P(18,3)*P(12,2) + P(12,3)*P(18,2) = 646,272 + 403,920 = 1,050,192.Therefore, the answer to the second question is 1,050,192.So, summarizing:1. The number of sequences with at least 2 organic and 2 biodynamic wines is 10,501,920.2. The number of sequences with at least 2 of each and no two consecutive same type is 1,050,192.I think that's it. Let me just double-check the calculations to make sure I didn't make any arithmetic errors.For the first question:C(18,2) = 153, C(12,3) = 220, so 153*220 = 33,660.C(18,3) = 816, C(12,2) = 66, so 816*66 = 53,856.Total combinations: 33,660 + 53,856 = 87,516.Multiply by 5! = 120: 87,516 * 120 = 10,501,920. Correct.For the second question:P(18,3) = 4,896, P(12,2) = 132, so 4,896*132 = 646,272.P(12,3) = 1,320, P(18,2) = 306, so 1,320*306 = 403,920.Total: 646,272 + 403,920 = 1,050,192. Correct.Yes, looks good.</think>

🧠:<think>Alright, so I have this problem where Alex, an organic farmer, wants to maximize his revenue by selling his produce in three different markets: A, B, and C. Each market has its own demand function, and Alex has a total of 150 units to allocate. I need to figure out how much to sell in each market and at what price to maximize the total revenue. Hmm, okay, let's break this down step by step.First, let's write down the given information. The demand functions for each market are:- Market A: ( q_A = 120 - 2p_A )- Market B: ( q_B = 100 - p_B )- Market C: ( q_C = 150 - 3p_C )And the total quantity sold across all markets can't exceed 150 units. So, ( q_A + q_B + q_C leq 150 ). But since Alex wants to maximize revenue, I think he would want to sell as much as possible, so probably the total will be exactly 150 units. That makes sense because selling more would mean higher revenue, assuming prices are set optimally.Revenue is calculated as the sum of revenues from each market, so:( R = p_A q_A + p_B q_B + p_C q_C )Our goal is to maximize R subject to the constraint ( q_A + q_B + q_C = 150 ).Since we have three variables here: ( p_A, p_B, p_C ), and each ( q ) is a function of their respective ( p ), we can substitute the demand functions into the revenue equation. That way, revenue will be expressed solely in terms of the prices, and we can take derivatives to find the maximum.So, let's substitute each ( q ):( R = p_A (120 - 2p_A) + p_B (100 - p_B) + p_C (150 - 3p_C) )Let me expand this:( R = 120p_A - 2p_A^2 + 100p_B - p_B^2 + 150p_C - 3p_C^2 )Now, we also have the constraint that the total quantity sold is 150:( (120 - 2p_A) + (100 - p_B) + (150 - 3p_C) = 150 )Let me simplify this constraint equation:First, add up the constants: 120 + 100 + 150 = 370Then, the terms with p: -2p_A - p_B - 3p_CSo, the equation becomes:370 - 2p_A - p_B - 3p_C = 150Subtract 150 from both sides:220 - 2p_A - p_B - 3p_C = 0Which simplifies to:2p_A + p_B + 3p_C = 220So, now we have our constraint: 2p_A + p_B + 3p_C = 220Now, to maximize R, we can use the method of Lagrange multipliers because we have a function to maximize (R) subject to a constraint. Alternatively, since we have three variables and one equation, we can express two variables in terms of the third and substitute back into R. But Lagrange multipliers might be more straightforward here.Let me recall how Lagrange multipliers work. If we have a function to maximize, say f(p_A, p_B, p_C), subject to a constraint g(p_A, p_B, p_C) = 0, then we set up the Lagrangian:( mathcal{L} = f - lambda g )Then, take partial derivatives with respect to each variable and set them equal to zero.So, in our case, f is the revenue function:( f = 120p_A - 2p_A^2 + 100p_B - p_B^2 + 150p_C - 3p_C^2 )And the constraint g is:( g = 2p_A + p_B + 3p_C - 220 = 0 )So, the Lagrangian is:( mathcal{L} = 120p_A - 2p_A^2 + 100p_B - p_B^2 + 150p_C - 3p_C^2 - lambda(2p_A + p_B + 3p_C - 220) )Now, we need to take partial derivatives of ( mathcal{L} ) with respect to p_A, p_B, p_C, and λ, and set each to zero.Let's compute each partial derivative:1. Partial derivative with respect to p_A:( frac{partial mathcal{L}}{partial p_A} = 120 - 4p_A - 2lambda = 0 )2. Partial derivative with respect to p_B:( frac{partial mathcal{L}}{partial p_B} = 100 - 2p_B - lambda = 0 )3. Partial derivative with respect to p_C:( frac{partial mathcal{L}}{partial p_C} = 150 - 6p_C - 3lambda = 0 )4. Partial derivative with respect to λ:( frac{partial mathcal{L}}{partial lambda} = -(2p_A + p_B + 3p_C - 220) = 0 )So, we have four equations:1. ( 120 - 4p_A - 2lambda = 0 ) --> Let's call this Equation (1)2. ( 100 - 2p_B - lambda = 0 ) --> Equation (2)3. ( 150 - 6p_C - 3lambda = 0 ) --> Equation (3)4. ( 2p_A + p_B + 3p_C = 220 ) --> Equation (4)Now, let's solve these equations step by step.From Equation (2): ( 100 - 2p_B - lambda = 0 )We can express λ in terms of p_B:( lambda = 100 - 2p_B ) --> Equation (2a)Similarly, from Equation (1): ( 120 - 4p_A - 2lambda = 0 )We can plug λ from Equation (2a) into Equation (1):( 120 - 4p_A - 2(100 - 2p_B) = 0 )Let me compute this:120 - 4p_A - 200 + 4p_B = 0Combine constants: 120 - 200 = -80So, -80 -4p_A + 4p_B = 0Let's rearrange:-4p_A + 4p_B = 80Divide both sides by 4:-p_A + p_B = 20So, p_B = p_A + 20 --> Equation (1a)Similarly, from Equation (3): ( 150 - 6p_C - 3lambda = 0 )Again, substitute λ from Equation (2a):( 150 - 6p_C - 3(100 - 2p_B) = 0 )Compute this:150 - 6p_C - 300 + 6p_B = 0Combine constants: 150 - 300 = -150So, -150 -6p_C + 6p_B = 0Rearrange:-6p_C + 6p_B = 150Divide both sides by 6:-p_C + p_B = 25So, p_B = p_C + 25 --> Equation (3a)Now, from Equation (1a): p_B = p_A + 20From Equation (3a): p_B = p_C + 25So, set them equal:p_A + 20 = p_C + 25Therefore, p_A = p_C + 5 --> Equation (5)So, now we have p_A in terms of p_C, and p_B in terms of p_C.Let me write down:p_A = p_C + 5p_B = p_C + 25Now, let's substitute these into Equation (4): 2p_A + p_B + 3p_C = 220Substitute p_A and p_B:2(p_C + 5) + (p_C + 25) + 3p_C = 220Let me expand this:2p_C + 10 + p_C + 25 + 3p_C = 220Combine like terms:2p_C + p_C + 3p_C = 6p_C10 + 25 = 35So, 6p_C + 35 = 220Subtract 35:6p_C = 185Divide by 6:p_C = 185 / 6 ≈ 30.8333So, p_C is approximately 30.8333.Now, let's find p_A and p_B.From Equation (5): p_A = p_C + 5 = 30.8333 + 5 = 35.8333From Equation (3a): p_B = p_C + 25 = 30.8333 + 25 = 55.8333So, p_A ≈ 35.83, p_B ≈ 55.83, p_C ≈ 30.83Now, let's compute the quantities sold in each market.From Market A: q_A = 120 - 2p_A = 120 - 2*(35.8333) = 120 - 71.6666 ≈ 48.3334From Market B: q_B = 100 - p_B = 100 - 55.8333 ≈ 44.1667From Market C: q_C = 150 - 3p_C = 150 - 3*(30.8333) = 150 - 92.5 ≈ 57.5Let me check if these quantities add up to 150:48.3334 + 44.1667 + 57.5 ≈ 48.3334 + 44.1667 = 92.5; 92.5 + 57.5 = 150. Perfect.Now, let's compute the revenue:R = p_A q_A + p_B q_B + p_C q_CCompute each term:p_A q_A = 35.8333 * 48.3334 ≈ Let's compute 35.8333 * 48.3334.First, 35 * 48 = 1680Then, 0.8333 * 48 ≈ 4035 * 0.3334 ≈ 11.6690.8333 * 0.3334 ≈ 0.2778So, adding all together:1680 + 40 + 11.669 + 0.2778 ≈ 1731.9468Wait, that seems a bit messy. Maybe a better way is to compute 35.8333 * 48.3334.Alternatively, since 35.8333 is 35 + 5/6, and 48.3334 is 48 + 1/3.So, 35.8333 * 48.3334 = (35 + 5/6)(48 + 1/3)Multiply 35*48 = 168035*(1/3) = 35/3 ≈ 11.6667(5/6)*48 = 40(5/6)*(1/3) = 5/18 ≈ 0.2778So, total is 1680 + 11.6667 + 40 + 0.2778 ≈ 1680 + 51.9445 ≈ 1731.9445Similarly, p_B q_B = 55.8333 * 44.1667Again, 55.8333 is 55 + 5/6, 44.1667 is 44 + 1/6.Multiply (55 + 5/6)(44 + 1/6)55*44 = 242055*(1/6) ≈ 9.1667(5/6)*44 ≈ 36.6667(5/6)*(1/6) ≈ 0.1389So, total is 2420 + 9.1667 + 36.6667 + 0.1389 ≈ 2420 + 46 ≈ 2466Wait, let's compute it more accurately:55.8333 * 44.1667Compute 55 * 44 = 242055 * 0.1667 ≈ 9.16850.8333 * 44 ≈ 36.66520.8333 * 0.1667 ≈ 0.1389So, adding all together:2420 + 9.1685 + 36.6652 + 0.1389 ≈ 2420 + 46 ≈ 2466Wait, but 9.1685 + 36.6652 = 45.8337, plus 0.1389 is 45.9726, so total is 2420 + 45.9726 ≈ 2465.9726Similarly, p_C q_C = 30.8333 * 57.5Compute 30 * 57.5 = 17250.8333 * 57.5 ≈ 47.9163So, total is 1725 + 47.9163 ≈ 1772.9163Now, sum up all three revenues:p_A q_A ≈ 1731.9445p_B q_B ≈ 2465.9726p_C q_C ≈ 1772.9163Total R ≈ 1731.9445 + 2465.9726 + 1772.9163Compute step by step:1731.9445 + 2465.9726 ≈ 4197.91714197.9171 + 1772.9163 ≈ 5970.8334So, total revenue is approximately 5970.83 dollars.Wait, that seems quite high. Let me double-check my calculations because I might have made an error in the multiplication.Wait, 35.8333 * 48.3334: Let's compute 35.8333 * 48.3334 more accurately.35.8333 * 48.3334Let me compute 35.8333 * 48 = 35.8333 * 40 + 35.8333 * 835.8333 * 40 = 1433.33235.8333 * 8 = 286.6664So, total is 1433.332 + 286.6664 ≈ 1719.9984Then, 35.8333 * 0.3334 ≈ 11.9444So, total is 1719.9984 + 11.9444 ≈ 1731.9428Similarly, p_B q_B = 55.8333 * 44.1667Compute 55.8333 * 44 = 55.8333 * 40 + 55.8333 * 455.8333 * 40 = 2233.33255.8333 * 4 = 223.3332Total: 2233.332 + 223.3332 ≈ 2456.6652Then, 55.8333 * 0.1667 ≈ 9.2708So, total is 2456.6652 + 9.2708 ≈ 2465.936p_C q_C = 30.8333 * 57.5Compute 30 * 57.5 = 17250.8333 * 57.5 ≈ 47.9163Total: 1725 + 47.9163 ≈ 1772.9163Now, adding all together:1731.9428 + 2465.936 + 1772.9163First, 1731.9428 + 2465.936 ≈ 4197.8788Then, 4197.8788 + 1772.9163 ≈ 5970.7951So, approximately 5970.80 dollars.Wait, but let me think, is this the maximum revenue? It seems high, but given the quantities and prices, maybe it's correct. Let me check if the prices are set correctly.From the demand functions:For Market A: q_A = 120 - 2p_AIf p_A ≈ 35.83, then q_A ≈ 120 - 71.66 ≈ 48.34, which matches.For Market B: q_B = 100 - p_B ≈ 100 - 55.83 ≈ 44.17, which matches.For Market C: q_C = 150 - 3p_C ≈ 150 - 92.5 ≈ 57.5, which matches.So, the quantities are correct.But let me check if these prices are indeed the ones that maximize revenue. Maybe I should verify by checking the second derivative or the bordered Hessian, but since this is a constrained optimization, the second derivative test is a bit more involved.Alternatively, since we used Lagrange multipliers and found a critical point, and given that the revenue function is quadratic and concave (since the coefficients of p_A^2, p_B^2, p_C^2 are negative), the critical point should be a maximum.Therefore, the solution we found is indeed the maximum.So, to recap:Optimal prices:p_A ≈ 35.83p_B ≈ 55.83p_C ≈ 30.83Optimal quantities:q_A ≈ 48.33q_B ≈ 44.17q_C ≈ 57.5Total revenue ≈ 5970.80But let me express these in fractions instead of decimals for precision.From earlier, p_C = 185/6 ≈ 30.8333So, p_C = 185/6Then, p_A = p_C + 5 = 185/6 + 30/6 = 215/6 ≈ 35.8333p_B = p_C + 25 = 185/6 + 150/6 = 335/6 ≈ 55.8333So, in fractions:p_A = 215/6p_B = 335/6p_C = 185/6Quantities:q_A = 120 - 2p_A = 120 - 2*(215/6) = 120 - 430/6 = 120 - 71.6667 = 48.3333 = 145/3Wait, 120 is 720/6, so 720/6 - 430/6 = 290/6 = 145/3 ≈ 48.3333Similarly, q_B = 100 - p_B = 100 - 335/6 = 600/6 - 335/6 = 265/6 ≈ 44.1667q_C = 150 - 3p_C = 150 - 3*(185/6) = 150 - 555/6 = 150 - 92.5 = 57.5 = 115/2So, in fractions:q_A = 145/3q_B = 265/6q_C = 115/2Let me verify the total quantity:145/3 + 265/6 + 115/2Convert all to sixths:145/3 = 290/6265/6 = 265/6115/2 = 345/6Total: 290 + 265 + 345 = 900/6 = 150. Perfect.So, all quantities add up correctly.Therefore, the optimal solution is:p_A = 215/6 ≈ 35.83p_B = 335/6 ≈ 55.83p_C = 185/6 ≈ 30.83q_A = 145/3 ≈ 48.33q_B = 265/6 ≈ 44.17q_C = 115/2 = 57.5Total revenue R = 5970.80 dollars approximately.But let me compute R exactly using fractions.R = p_A q_A + p_B q_B + p_C q_CCompute each term:p_A q_A = (215/6) * (145/3) = (215*145)/(6*3) = (31175)/18p_B q_B = (335/6) * (265/6) = (335*265)/(6*6) = (88,775)/36p_C q_C = (185/6) * (115/2) = (185*115)/(6*2) = (21,275)/12Now, let's compute each fraction:31175/18 ≈ 1731.944488775/36 ≈ 2465.972221275/12 ≈ 1772.9167Adding them together:1731.9444 + 2465.9722 + 1772.9167 ≈ 5970.8333So, exactly, R = 5970 + 5/6 ≈ 5970.8333 dollars.Therefore, the exact total revenue is 5970 and 5/6 dollars, which is approximately 5970.83 dollars.So, summarizing:Optimal prices:- Market A: 215/6 ≈ 35.83 dollars- Market B: 335/6 ≈ 55.83 dollars- Market C: 185/6 ≈ 30.83 dollarsOptimal quantities:- Market A: 145/3 ≈ 48.33 units- Market B: 265/6 ≈ 44.17 units- Market C: 115/2 = 57.5 unitsTotal revenue: 5970 5/6 dollars ≈ 5970.83 dollarsI think this makes sense. Each market's price is set such that the marginal revenue from each market is equal, considering the constraint on total quantity. The higher prices in Markets A and B reflect their higher demand elasticity, while Market C has a lower price due to its more elastic demand. The quantities allocated reflect this as well, with Market C selling the most units, followed by Market A, then Market B.Just to make sure, let me check the marginal revenues.Marginal revenue for each market is the derivative of revenue with respect to quantity, which is equal to price minus the change in price due to the change in quantity.But since we used Lagrange multipliers, the condition is that the marginal revenue from each market should be equal, adjusted by the constraint. However, in this case, the Lagrange multiplier method already accounts for the constraint, so the critical point we found should satisfy the necessary conditions for maximum revenue.Therefore, I'm confident that the solution is correct.</think>

🧠:<think>Alright, so I've got these two probability questions to solve. Let me take them one at a time and think through each step carefully.Problem 1: Alex visits 10 stores, each with a 20% chance of having a sale. I need to find the probability that at least 3 stores have a sale. Hmm, okay. So this sounds like a binomial probability problem because each store is an independent trial with two outcomes: sale or no sale.First, let me recall the binomial probability formula. The probability of exactly k successes in n trials is given by:[ P(k) = C(n, k) times p^k times (1-p)^{n-k} ]Where ( C(n, k) ) is the combination of n things taken k at a time, p is the probability of success, and ( 1-p ) is the probability of failure.In this case, n = 10, p = 0.2, and we want the probability of at least 3 successes, which is P(3) + P(4) + ... + P(10). But calculating each of these individually might be tedious. Instead, it's often easier to calculate the complement probability and subtract it from 1.The complement of "at least 3" is "fewer than 3," which is P(0) + P(1) + P(2). So, let's compute that.First, let's compute P(0):[ P(0) = C(10, 0) times (0.2)^0 times (0.8)^{10} ][ C(10, 0) = 1 ][ (0.2)^0 = 1 ][ (0.8)^{10} approx 0.1073741824 ]So, P(0) ≈ 0.1073741824Next, P(1):[ P(1) = C(10, 1) times (0.2)^1 times (0.8)^9 ][ C(10, 1) = 10 ][ (0.2)^1 = 0.2 ][ (0.8)^9 ≈ 0.134217728 ]So, P(1) ≈ 10 * 0.2 * 0.134217728 ≈ 10 * 0.0268435456 ≈ 0.268435456Now, P(2):[ P(2) = C(10, 2) times (0.2)^2 times (0.8)^8 ][ C(10, 2) = 45 ][ (0.2)^2 = 0.04 ][ (0.8)^8 ≈ 0.16777216 ]So, P(2) ≈ 45 * 0.04 * 0.16777216 ≈ 45 * 0.0067108864 ≈ 0.301989888Adding these up:P(0) + P(1) + P(2) ≈ 0.1073741824 + 0.268435456 + 0.301989888 ≈ 0.6777995264Therefore, the probability of at least 3 sales is:1 - 0.6777995264 ≈ 0.3222004736So, approximately 32.22%.Wait, let me double-check my calculations to make sure I didn't make any arithmetic errors.Calculating P(0):(0.8)^10 is indeed approximately 0.1073741824.P(1):C(10,1) is 10, 0.2 * 0.8^9. 0.8^9 is 0.134217728, so 10 * 0.2 * 0.134217728 = 10 * 0.0268435456 = 0.268435456. That seems right.P(2):C(10,2) is 45, 0.2^2 is 0.04, 0.8^8 is 0.16777216. Multiplying these together: 45 * 0.04 = 1.8; 1.8 * 0.16777216 ≈ 0.301989888. Correct.Adding them up: 0.1073741824 + 0.268435456 = 0.3758096384; plus 0.301989888 gives 0.6777995264. So 1 - 0.6777995264 is indeed approximately 0.3222004736, which is about 32.22%.So, I think that's correct.Problem 2: Over 12 visits, Alex tracks the number of stores with sales. We need the expected value and variance of the total number of stores with sales.Hmm, okay. So each visit is a binomial trial with n=10, p=0.2. So, per visit, the number of sales is a binomial random variable with parameters n=10, p=0.2.But over 12 visits, the total number of sales would be the sum of 12 independent binomial variables, each with n=10, p=0.2. Alternatively, this is equivalent to a binomial distribution with n=120 (since 12*10) and p=0.2.Wait, is that correct? Let me think.Each visit is 10 trials, so over 12 visits, it's 12*10=120 trials, each with p=0.2. So yes, the total number of sales is binomial with n=120, p=0.2.Therefore, the expected value E[X] is n*p, which is 120*0.2 = 24.The variance Var(X) is n*p*(1-p) = 120*0.2*0.8 = 120*0.16 = 19.2.Alternatively, since each visit is a binomial variable with E[X_i] = 10*0.2=2 and Var(X_i)=10*0.2*0.8=1.6. Then, over 12 visits, the total expectation is 12*2=24, and the total variance is 12*1.6=19.2, since variances add for independent variables.Either way, same result.So, the expected value is 24, variance is 19.2.Wait, let me make sure I didn't confuse anything here. So, each visit is 10 stores, each with 20% chance. So, per visit, expected number of sales is 10*0.2=2, variance is 10*0.2*0.8=1.6. Then, over 12 visits, since each visit is independent, the total expectation is 12*2=24, and the total variance is 12*1.6=19.2. That seems correct.Alternatively, if we model it as 120 independent trials, each with p=0.2, then E[X] = 120*0.2=24, Var(X)=120*0.2*0.8=19.2. Same result.So, I think that's solid.Summary:1. The probability that at least 3 out of 10 stores have a sale is approximately 32.22%.2. Over 12 visits, the expected number of sales is 24, and the variance is 19.2.I think that's it. Let me just write the final answers in the required format.Final Answer1. The probability is boxed{0.3222}.2. The expected value is boxed{24} and the variance is boxed{19.2}.</think>