💡Okay, so I have this matrix B, which is a 2x2 matrix:B = [[3, 4], [0, 2]]And I need to find B^15 minus 3 times B^14. Hmm, that seems like a lot of matrix multiplication, but maybe there's a pattern or a smarter way to do this instead of multiplying the matrix 14 or 15 times.Let me think. I remember that for matrices, sometimes you can use eigenvalues or diagonalization to simplify powers of matrices. But I'm not sure if B is diagonalizable. Alternatively, maybe I can find a recurrence relation or use the characteristic equation.First, let me try to compute B - 3I, where I is the identity matrix. That might help because subtracting 3I could simplify things.So, B - 3I is:[[3-3, 4], [0, 2-3]] = [[0, 4], [0, -1]]Alright, that's a simpler matrix. Let me call this matrix C for now. So, C = B - 3I = [[0, 4], [0, -1]]Now, maybe I can compute B*C to see if there's a pattern.B*C = [[3, 4], [0, 2]] * [[0, 4], [0, -1]]Let me compute that:First row, first column: 3*0 + 4*0 = 0First row, second column: 3*4 + 4*(-1) = 12 - 4 = 8Second row, first column: 0*0 + 2*0 = 0Second row, second column: 0*4 + 2*(-1) = -2So, B*C = [[0, 8], [0, -2]]Wait a minute, that looks like 2*C because 2*C would be [[0, 8], [0, -2]]. So, B*C = 2C.That's interesting. So, B*(B - 3I) = 2*(B - 3I). Maybe this can help me find a pattern for higher powers.Let me try to see what happens when I compute B^2*C.B^2*C = B*(B*C) = B*(2C) = 2*(B*C) = 2*(2C) = 4CSimilarly, B^3*C = B*(B^2*C) = B*(4C) = 4*(B*C) = 4*(2C) = 8CI see a pattern here. It seems that B^n*C = 2^{n-1}*C for n >= 1.Wait, let me check for n=1: B^1*C = B*C = 2C, which is 2^{1-1}*C = 1*C, but that's not matching. Hmm, maybe n >= 2?Wait, for n=2: B^2*C = 4C = 2^{2-1}*C = 2C? No, that's not matching either. Wait, 4C is 2^2*C, so maybe B^n*C = 2^{n-1}*C?Wait, for n=1: B^1*C = 2C = 2^{1}*C, so maybe B^n*C = 2^{n}*C?Wait, let me recast that.From B*C = 2C, so B*C = 2C.Then, B^2*C = B*(B*C) = B*(2C) = 2*(B*C) = 2*(2C) = 4C = 2^2*CSimilarly, B^3*C = B*(B^2*C) = B*(4C) = 4*(B*C) = 4*(2C) = 8C = 2^3*CSo, in general, B^n*C = 2^n*CWait, but when n=1, B^1*C = 2C = 2^1*C, which works. So, the pattern is B^n*C = 2^n*C.So, for any n >=1, B^n*(B - 3I) = 2^n*(B - 3I)Wait, but in the problem, I have B^15 - 3B^14.Let me factor that as B^14*(B - 3I). So, B^15 - 3B^14 = B^14*(B - 3I)From the pattern above, B^n*(B - 3I) = 2^n*(B - 3I). So, for n=14, B^14*(B - 3I) = 2^{14}*(B - 3I)Therefore, B^15 - 3B^14 = 2^{14}*(B - 3I)Now, let's compute 2^{14}*(B - 3I). We already found that B - 3I is [[0,4],[0,-1]]So, 2^{14} is 16384.Wait, no, 2^10 is 1024, 2^14 is 16384? Wait, 2^10 is 1024, 2^11 is 2048, 2^12 is 4096, 2^13 is 8192, 2^14 is 16384.Wait, but in the initial problem, the assistant's answer was 2^{13}*(B - 3I). Hmm, maybe I made a mistake.Wait, let me check again.We have B^n*(B - 3I) = 2^n*(B - 3I). So, for n=14, it's 2^{14}*(B - 3I). So, the result should be 2^{14} times [[0,4],[0,-1]]Which would be [[0, 4*16384], [0, -1*16384]] = [[0, 65536], [0, -16384]]But the assistant's answer was [[0, 8192], [0, -8192]], which is 2^{13}*(B - 3I). So, perhaps I made a mistake in the exponent.Wait, let me go back.We have B*C = 2C, so B*C = 2C.Then, B^2*C = B*(B*C) = B*(2C) = 2*(B*C) = 2*(2C) = 4C = 2^2*CSimilarly, B^3*C = 8C = 2^3*CSo, in general, B^n*C = 2^n*CTherefore, B^14*C = 2^{14}*CBut in the problem, we have B^15 - 3B^14 = B^14*(B - 3I) = B^14*CWhich is 2^{14}*CSo, 2^{14} is 16384, so 16384*C is [[0, 4*16384], [0, -1*16384]] = [[0, 65536], [0, -16384]]But the assistant's answer was [[0, 8192], [0, -8192]], which is 2^{13}*C.Wait, perhaps I made a mistake in the pattern. Let me check again.Wait, when n=1: B^1*C = 2C = 2^1*Cn=2: B^2*C = 4C = 2^2*Cn=3: B^3*C = 8C = 2^3*CSo, in general, B^n*C = 2^n*CTherefore, for n=14, B^14*C = 2^{14}*CSo, 2^{14} is 16384, so 16384*C is [[0, 4*16384], [0, -1*16384]] = [[0, 65536], [0, -16384]]But the assistant's answer was [[0, 8192], [0, -8192]], which is 2^{13}*C.Hmm, maybe the assistant made a mistake in the exponent.Wait, let me see. The assistant wrote:"Thus, B^{15} - 3 B^{14} = B^{14} (B - 3 I) = 2^{13} (B - 3 I) = 2^{13} [[0,4],[0,-1]] = [[0,8192],[0,-8192]]"But according to my calculation, it should be 2^{14}*(B - 3I), which would be 16384*C.Wait, perhaps the assistant made a mistake in the exponent. Let me check the steps again.The assistant wrote:"Next, we compute B (B - 3 I):B (B - 3 I) = [[3,4],[0,2]] * [[0,4],[0,-1]] = [[0,8],[0,-2]] = 2 (B - 3 I)."Yes, that's correct. So, B*(B - 3I) = 2*(B - 3I)Then, the assistant says:"Now, for any n >= 2,B^n (B - 3 I) = 2^{n-1} (B - 3 I),we haveB^{15} (B - 3 I) = 2^{14} (B - 3 I)."Wait, so the assistant is saying that B^n*(B - 3I) = 2^{n-1}*(B - 3I). But from our earlier calculation, B^n*(B - 3I) = 2^n*(B - 3I). So, the assistant is off by one in the exponent.Therefore, the correct result should be 2^{14}*(B - 3I), which is 16384*C.So, the assistant's answer is incorrect because they used 2^{13} instead of 2^{14}.Therefore, the correct answer should be:[[0, 4*16384], [0, -1*16384]] = [[0, 65536], [0, -16384]]But let me double-check.Wait, let's compute B^1*C = 2CB^2*C = 2^2*C...B^n*C = 2^n*CSo, B^14*C = 2^{14}*CTherefore, B^15 - 3B^14 = B^14*(B - 3I) = B^14*C = 2^{14}*CSo, 2^{14} is 16384, so 16384*C is [[0, 4*16384], [0, -1*16384]] = [[0, 65536], [0, -16384]]But the assistant's answer was 2^{13}*C, which is 8192*C, giving [[0, 32768], [0, -8192]]Wait, no, 4*8192 is 32768, but the assistant wrote 8192 in the second entry. Wait, let me check:Wait, 2^{13} is 8192, so 8192*C is [[0, 4*8192], [0, -1*8192]] = [[0, 32768], [0, -8192]]But the assistant wrote [[0, 8192], [0, -8192]], which is 8192*C, but that would be 2^{13}*C.Wait, but 4*8192 is 32768, not 8192. So, the assistant made a mistake in scaling.Wait, perhaps the assistant thought that 2^{13}*(B - 3I) is 8192*(B - 3I), but actually, it's 8192*C, which would be [[0, 32768], [0, -8192]]But the assistant wrote [[0, 8192], [0, -8192]], which is incorrect because 4*8192 is 32768, not 8192.Wait, perhaps the assistant made a mistake in the scaling factor.Alternatively, maybe I made a mistake in interpreting the pattern.Wait, let me think differently. Maybe instead of using the pattern, I can diagonalize B or use its eigenvalues.Let me try to find the eigenvalues of B.The characteristic equation is det(B - λI) = 0So,|3 - λ 4 ||0 2 - λ | = (3 - λ)(2 - λ) = 0So, eigenvalues are λ = 3 and λ = 2.So, B is diagonalizable because it has distinct eigenvalues.Therefore, B can be written as PDP^{-1}, where D is the diagonal matrix of eigenvalues.So, D = [[3, 0], [0, 2]]Then, B^n = PD^nP^{-1}So, if I can find P and P^{-1}, I can compute B^n easily.Let me find the eigenvectors for each eigenvalue.For λ = 3:(B - 3I)v = 0So,[[0, 4], [0, -1]] * [v1; v2] = [0; 0]From the second row: -v2 = 0 => v2 = 0From the first row: 4v2 = 0, which is already satisfied.So, the eigenvector is [1; 0]For λ = 2:(B - 2I)v = 0[[1, 4], [0, 0]] * [v1; v2] = [0; 0]From the first row: v1 + 4v2 = 0 => v1 = -4v2So, the eigenvector can be [ -4; 1 ]Therefore, matrix P is:P = [[1, -4], [0, 1]]And P^{-1} is:Since P is upper triangular with 1s on the diagonal, its inverse is also upper triangular with 1s on the diagonal, and the off-diagonal element is the negative of the original.So, P^{-1} = [[1, 4], [0, 1]]Let me verify:P * P^{-1} = [[1*1 + (-4)*0, 1*4 + (-4)*1], [0*1 + 1*0, 0*4 + 1*1]] = [[1, 0], [0, 1]]Yes, correct.So, now, B^n = P D^n P^{-1}Where D^n = [[3^n, 0], [0, 2^n]]So, let's compute B^n:B^n = P * D^n * P^{-1} = [[1, -4], [0, 1]] * [[3^n, 0], [0, 2^n]] * [[1, 4], [0, 1]]Let me compute this step by step.First, compute P * D^n:[[1*3^n + (-4)*0, 1*0 + (-4)*2^n], [0*3^n + 1*0, 0*0 + 1*2^n]]Which simplifies to:[[3^n, -4*2^n], [0, 2^n]]Now, multiply this by P^{-1}:[[3^n, -4*2^n], [0, 2^n]] * [[1, 4], [0, 1]]Compute each element:First row, first column: 3^n*1 + (-4*2^n)*0 = 3^nFirst row, second column: 3^n*4 + (-4*2^n)*1 = 4*3^n - 4*2^nSecond row, first column: 0*1 + 2^n*0 = 0Second row, second column: 0*4 + 2^n*1 = 2^nSo, B^n = [[3^n, 4*3^n - 4*2^n], [0, 2^n]]Therefore, B^15 = [[3^15, 4*3^15 - 4*2^15], [0, 2^15]]Similarly, B^14 = [[3^14, 4*3^14 - 4*2^14], [0, 2^14]]Now, compute B^15 - 3B^14:Compute each element:First row, first column: 3^15 - 3*3^14 = 3^15 - 3^15 = 0First row, second column: (4*3^15 - 4*2^15) - 3*(4*3^14 - 4*2^14) = 4*3^15 - 4*2^15 - 12*3^14 + 12*2^14Simplify:Factor out 4*3^14: 4*3^14*(3 - 3) = 0Wait, let me compute each term:4*3^15 = 4*3*3^14 = 12*3^14-4*2^15 = -4*2*2^14 = -8*2^14-12*3^14+12*2^14So, combining:12*3^14 - 12*3^14 = 0-8*2^14 +12*2^14 = 4*2^14So, first row, second column: 4*2^14Second row, first column: 0 - 3*0 = 0Second row, second column: 2^15 - 3*2^14 = 2^14*(2 - 3) = -2^14So, putting it all together:B^15 - 3B^14 = [[0, 4*2^14], [0, -2^14]]Compute 2^14: 16384So, 4*16384 = 65536Thus, the matrix is:[[0, 65536], [0, -16384]]Wait, but earlier when I used the pattern, I thought it should be 2^{14}*C, which is 16384*C = [[0, 65536], [0, -16384]]So, that's consistent with the diagonalization method.But the assistant's answer was [[0, 8192], [0, -8192]], which is 2^{13}*C.So, the assistant made a mistake in the exponent, using 2^{13} instead of 2^{14}.Therefore, the correct answer should be [[0, 65536], [0, -16384]]But wait, let me check the calculation again.Wait, in the diagonalization approach, I got B^15 - 3B^14 = [[0, 4*2^14], [0, -2^14]]Which is [[0, 4*16384], [0, -16384]] = [[0, 65536], [0, -16384]]But the assistant's answer was [[0, 8192], [0, -8192]], which is 2^{13}*C.So, perhaps the assistant made a mistake in the exponent, using n=13 instead of n=14.Alternatively, maybe I made a mistake in the diagonalization approach.Wait, let me check the diagonalization steps again.We had B^n = [[3^n, 4*3^n - 4*2^n], [0, 2^n]]Then, B^15 - 3B^14:First row, first column: 3^15 - 3*3^14 = 3^15 - 3^15 = 0First row, second column: (4*3^15 - 4*2^15) - 3*(4*3^14 - 4*2^14) = 4*3^15 - 4*2^15 - 12*3^14 + 12*2^14Factor:4*3^15 - 12*3^14 = 4*3^14*(3 - 3) = 0-4*2^15 +12*2^14 = -4*2*2^14 +12*2^14 = (-8 +12)*2^14 = 4*2^14So, first row, second column: 4*2^14 = 4*16384 = 65536Second row, second column: 2^15 - 3*2^14 = 2*2^14 - 3*2^14 = (2 -3)*2^14 = -2^14 = -16384So, yes, the result is [[0, 65536], [0, -16384]]Therefore, the correct answer is [[0, 65536], [0, -16384]]But the assistant's answer was [[0, 8192], [0, -8192]], which is incorrect.So, in conclusion, the correct answer is [[0, 65536], [0, -16384]]

💡Alright, so I've got this problem here about permutations of the numbers 1 through n, and I need to find how many of these permutations satisfy the condition that for every position k (from 1 to n), the number in that position, a_k, is at least k minus 2. That is, a_k ≥ k - 2 for all k.Hmm, okay. Let me try to unpack this. First, permutations of 1 through n are just all the possible ways to arrange these numbers. There are n! of them. But now, we have this condition that each number in the permutation has to be at least k - 2 in position k.Let me think about what this condition means for each position. For k = 1, the condition becomes a_1 ≥ 1 - 2, which simplifies to a_1 ≥ -1. But since all our numbers are positive integers starting from 1, this condition is automatically satisfied for k = 1 because the smallest a_1 can be is 1, which is greater than -1. So, no restrictions here.For k = 2, the condition is a_2 ≥ 2 - 2, which is a_2 ≥ 0. Again, since all numbers are at least 1, this is also automatically satisfied. So, no restrictions for k = 2 either.Starting from k = 3, things get interesting. For k = 3, a_3 ≥ 3 - 2, so a_3 ≥ 1. Well, that's still automatically true because the numbers are at least 1. So, no restriction here either.Wait, maybe I need to go further. Let's see, for k = 4, a_4 ≥ 4 - 2, which is a_4 ≥ 2. So, in position 4, the number can't be 1. It has to be 2 or higher. That's a restriction.Similarly, for k = 5, a_5 ≥ 5 - 2, so a_5 ≥ 3. So, in position 5, the number can't be 1 or 2; it has to be 3 or higher.Continuing this pattern, for position k, the number a_k must be at least k - 2. So, for k = 6, a_6 ≥ 4, and so on.Wait, so for each position k, the number a_k can't be less than k - 2. So, for each k, the number of forbidden choices is (k - 3) because numbers less than k - 2 are forbidden. But wait, for k = 3, forbidden numbers would be less than 1, which is none. For k = 4, forbidden numbers are less than 2, which is just 1. For k = 5, forbidden numbers are less than 3, which are 1 and 2, etc.So, in general, for each position k, the number of forbidden choices is (k - 3). But since we can't have negative forbidden choices, for k = 1 and 2, there are no forbidden choices, and starting from k = 3, the forbidden choices start at 1 and increase by 1 each time.But how does this affect the number of permutations? I need to count the number of permutations where each a_k is at least k - 2.Maybe it's easier to think about this recursively or using some combinatorial argument.Let me consider small values of n to see if I can spot a pattern.For n = 2:We have permutations of [1, 2]. The conditions are:- For k = 1: a_1 ≥ -1 (always true)- For k = 2: a_2 ≥ 0 (always true)So, all 2! = 2 permutations satisfy the condition. So, the answer for n = 2 is 2.For n = 3:Permutations are [1,2,3], [1,3,2], [2,1,3], [2,3,1], [3,1,2], [3,2,1].Conditions:- For k = 1: a_1 ≥ -1 (always true)- For k = 2: a_2 ≥ 0 (always true)- For k = 3: a_3 ≥ 1 (always true)So, all 6 permutations satisfy the condition. So, the answer for n = 3 is 6.Wait, that's 3! = 6, same as n!.Hmm, interesting. Let's try n = 4.For n = 4, the conditions are:- k = 1: a_1 ≥ -1 (always true)- k = 2: a_2 ≥ 0 (always true)- k = 3: a_3 ≥ 1 (always true)- k = 4: a_4 ≥ 2So, in position 4, the number can't be 1. So, a_4 must be 2, 3, or 4.So, how many permutations satisfy this?Total permutations: 24.Number of permutations where a_4 = 1: Let's fix a_4 = 1, then the remaining numbers 2, 3, 4 can be arranged freely in the first three positions. So, 3! = 6 permutations.Therefore, the number of valid permutations is 24 - 6 = 18.So, for n = 4, the answer is 18.Wait, 18 is equal to 2 * 3^(4-2) = 2 * 9 = 18. Interesting.Let me check for n = 5.For n = 5, the conditions are:- k = 1: a_1 ≥ -1 (always true)- k = 2: a_2 ≥ 0 (always true)- k = 3: a_3 ≥ 1 (always true)- k = 4: a_4 ≥ 2- k = 5: a_5 ≥ 3So, in position 4, a_4 can't be 1, and in position 5, a_5 can't be 1 or 2.So, how many permutations satisfy these conditions?Total permutations: 120.We need to subtract permutations where a_4 = 1 or a_5 = 1 or 2.But this might get complicated because of overlapping cases. Maybe inclusion-exclusion is needed.Alternatively, maybe we can model this as a permutation with restricted positions.Wait, another approach: For each position k, starting from k = 4, we have restrictions on the numbers that can be placed there.Perhaps we can model this as a recurrence relation.Wait, for n = 4, we had 18 permutations.If I think about n = 5, maybe the number of valid permutations is 18 * something.Wait, let's see:For n = 4, the number of valid permutations is 18.For n = 5, we need to place the number 5 somewhere, but also ensure that the other positions satisfy their respective conditions.Wait, maybe it's better to think recursively.Suppose we have a valid permutation for n - 1, and we want to insert n into the permutation for n.But I'm not sure if that's straightforward.Alternatively, let's think about the number of choices for each position.For position 1: Any number from 1 to n is allowed.For position 2: Any number from 1 to n is allowed.For position 3: Any number from 1 to n is allowed.For position 4: Numbers from 2 to n are allowed (since a_4 ≥ 2).For position 5: Numbers from 3 to n are allowed (since a_5 ≥ 3).Similarly, for position k, numbers from (k - 2) to n are allowed.Wait, but this is only for k ≥ 3.Wait, no, for k = 4, it's numbers from 2 to n.For k = 5, numbers from 3 to n.For k = 6, numbers from 4 to n, and so on.But this seems like a problem where each position has a certain number of allowed choices, but the choices are dependent because numbers can't repeat.So, it's not as simple as multiplying the number of choices for each position.Hmm, perhaps we can model this as a permutation with certain forbidden positions.Wait, another idea: The problem resembles derangements, but with different constraints.Alternatively, maybe we can model this as a permutation where each element is at least a certain distance from its original position.Wait, but in this case, the constraints are on the values, not on the positions relative to their original positions.Wait, perhaps we can think of it as a permutation where certain numbers are restricted from being in certain positions.For example, number 1 can't be in positions 4, 5, ..., n.Number 2 can't be in positions 5, 6, ..., n.Number 3 can't be in positions 6, 7, ..., n.And so on.So, for each number m, it can't be placed in positions k where k > m + 2.Wait, let's see:If a number m is placed in position k, then m ≥ k - 2.So, rearranged, k ≤ m + 2.Therefore, number m can only be placed in positions 1 through m + 2.So, for m = 1: can be placed in positions 1, 2, 3.For m = 2: can be placed in positions 1, 2, 3, 4.For m = 3: can be placed in positions 1, 2, 3, 4, 5.And so on, up to m = n: can be placed in any position.So, each number m has (m + 2) possible positions it can be placed in, except when m + 2 exceeds n, in which case it's just n positions.Wait, but for m = 1, m + 2 = 3, so positions 1, 2, 3.For m = 2, m + 2 = 4, so positions 1, 2, 3, 4.For m = 3, m + 2 = 5, so positions 1, 2, 3, 4, 5.And so on, until m = n - 2, which would have m + 2 = n, so positions 1 through n.For m = n - 1, m + 2 = n + 1, which is beyond n, so it can be placed in any position.Similarly, m = n can be placed in any position.So, each number m has (m + 2) possible positions, except when m + 2 > n, in which case it's n positions.But how does this help us count the number of permutations?This seems similar to counting the number of injective functions from the set of numbers to the set of positions, with each number having a restricted set of positions it can map to.This is similar to a problem in combinatorics called the "rook polynomial" problem, where you count the number of ways to place non-attacking rooks on a chessboard with certain forbidden positions.Alternatively, it's similar to counting the number of permutation matrices with certain forbidden positions.But I'm not sure about the exact formula for this.Wait, maybe we can model this as a permutation where each element has a certain number of available positions, and use the principle of inclusion-exclusion or recursion.Alternatively, perhaps we can think of it as a permutation where each number m can be placed in positions 1 through m + 2.So, for m = 1, positions 1, 2, 3.For m = 2, positions 1, 2, 3, 4.For m = 3, positions 1, 2, 3, 4, 5.And so on.So, the number of available positions increases as m increases.This seems like a problem that can be approached using recursion.Let me denote f(n) as the number of valid permutations for n elements.Now, let's try to find a recurrence relation for f(n).Consider the position of the number n in the permutation.Since n can be placed in any position from 1 to n, but we have to ensure that the permutation remains valid.Wait, but actually, n can be placed in any position, because for any position k, n ≥ k - 2 is always true since n ≥ k for all k ≤ n.So, placing n in any position is allowed.Therefore, for each valid permutation of n - 1 elements, we can insert n into any of the n positions, but we have to make sure that the rest of the permutation remains valid.Wait, but inserting n into a permutation of n - 1 elements might affect the validity of the permutation because the other numbers have their own constraints.Hmm, maybe this approach is not straightforward.Alternatively, let's think about the number of choices for each position, starting from the last position.For position n, a_n must be ≥ n - 2.So, a_n can be n - 2, n - 1, or n.So, there are 3 choices for a_n.Once we've chosen a_n, we move to position n - 1.For position n - 1, a_{n-1} must be ≥ (n - 1) - 2 = n - 3.So, a_{n-1} can be n - 3, n - 2, n - 1, or n, but we have to exclude the number already chosen for a_n.So, if a_n was n, then a_{n-1} can be n - 3, n - 2, or n - 1.If a_n was n - 1, then a_{n-1} can be n - 3, n - 2, or n.If a_n was n - 2, then a_{n-1} can be n - 3, n - 1, or n.Wait, so in each case, there are 3 choices for a_{n-1}.Similarly, for position n - 2, a_{n-2} must be ≥ (n - 2) - 2 = n - 4.So, a_{n-2} can be n - 4, n - 3, n - 2, n - 1, or n, but we have to exclude the numbers already chosen for a_n and a_{n-1}.But depending on what was chosen for a_n and a_{n-1}, the number of available choices might vary.Wait, but if we continue this pattern, it seems like for each position from n down to 3, there are 3 choices, and for positions 1 and 2, there are no restrictions.Wait, but that might not be the case because as we choose numbers for the higher positions, the available numbers for the lower positions decrease.Wait, let me try to formalize this.Suppose we start from position n.We have 3 choices for a_n: n - 2, n - 1, n.Then, for position n - 1, we have 3 choices as well, because one number has been used, but the forbidden number for position n - 1 is n - 3, so we have 4 possible numbers (n - 3, n - 2, n - 1, n), but one is already used, so 3 choices.Similarly, for position n - 2, a_{n-2} must be ≥ n - 4.So, the possible numbers are n - 4, n - 3, n - 2, n - 1, n.But two numbers have already been used in positions n and n - 1, so we have 5 - 2 = 3 choices.Wait, this seems to hold.So, for each position from n down to 3, we have 3 choices.For positions 1 and 2, we have no restrictions, so we can arrange the remaining numbers freely.Wait, but how many numbers are left for positions 1 and 2?After choosing numbers for positions 3 to n, which are n - 2 positions, we have used n - 2 numbers, leaving 2 numbers to be placed in positions 1 and 2.These two numbers can be arranged in 2! = 2 ways.So, putting it all together:For positions 3 to n: 3 choices each, so 3^{n - 2} possibilities.For positions 1 and 2: 2! = 2 possibilities.Therefore, the total number of valid permutations is 2 * 3^{n - 2}.Wait, let's check this with our earlier examples.For n = 2: 2 * 3^{0} = 2 * 1 = 2. Correct.For n = 3: 2 * 3^{1} = 6. Correct.For n = 4: 2 * 3^{2} = 18. Correct.For n = 5: 2 * 3^{3} = 54. Let's see if that's correct.Wait, for n = 5, total permutations are 120. If the number of valid permutations is 54, that seems plausible.But let me try to count it manually.For n = 5, the conditions are:- a_1: any- a_2: any- a_3: any- a_4: ≥ 2- a_5: ≥ 3So, a_4 can be 2, 3, 4, 5.a_5 can be 3, 4, 5.But we have to ensure that a_4 and a_5 are distinct.So, let's fix a_5 first.Case 1: a_5 = 3.Then, a_4 can be 2, 4, or 5.Subcase 1a: a_4 = 2.Then, the remaining numbers are 1, 4, 5.These need to be placed in positions 1, 2, 3.But there are no restrictions on positions 1, 2, 3, so 3! = 6 permutations.Subcase 1b: a_4 = 4.Then, the remaining numbers are 1, 2, 5.Again, 3! = 6 permutations.Subcase 1c: a_4 = 5.Then, the remaining numbers are 1, 2, 4.Again, 3! = 6 permutations.Total for Case 1: 6 + 6 + 6 = 18.Case 2: a_5 = 4.Then, a_4 can be 2, 3, or 5.Subcase 2a: a_4 = 2.Remaining numbers: 1, 3, 5.3! = 6.Subcase 2b: a_4 = 3.Remaining numbers: 1, 2, 5.3! = 6.Subcase 2c: a_4 = 5.Remaining numbers: 1, 2, 3.3! = 6.Total for Case 2: 18.Case 3: a_5 = 5.Then, a_4 can be 2, 3, or 4.Subcase 3a: a_4 = 2.Remaining numbers: 1, 3, 4.3! = 6.Subcase 3b: a_4 = 3.Remaining numbers: 1, 2, 4.3! = 6.Subcase 3c: a_4 = 4.Remaining numbers: 1, 2, 3.3! = 6.Total for Case 3: 18.So, total permutations: 18 + 18 + 18 = 54.Which matches 2 * 3^{5 - 2} = 2 * 27 = 54.So, this seems to hold.Therefore, the general formula is 2 * 3^{n - 2}.Hence, the number of distinct permutations satisfying the given condition is 2 * 3^{n - 2}.

💡Okay, so I have this problem about contestants in a math competition, and I need to prove that two specific contestants, A₁ and A₂, have the same number of friends. Let me try to break this down step by step.First, the problem says that among the n contestants, some know each other. If two contestants don't know each other, they have exactly two common friends. We're told that A₁ and A₂ know each other, but they don't have any common friends. I need to show that A₁ and A₂ have the same number of friends.Hmm, this sounds like a graph theory problem. Maybe I can model the contestants as vertices in a graph, and an edge between two vertices means those two contestants know each other. So, A₁ and A₂ are connected by an edge, and they don't share any common neighbors because they don't have any common friends.Let me denote the graph as G with vertices V = {A₁, A₂, ..., Aₙ}. The edges represent the acquaintance relationship. So, A₁ and A₂ are adjacent vertices, and their neighbor sets are disjoint, meaning N(A₁) ∩ N(A₂) = ∅.Now, the problem states that any two non-adjacent vertices have exactly two common neighbors. That seems like a specific property of the graph. Maybe it's a strongly regular graph or something similar? I'm not sure, but I can work with the given property.Since A₁ and A₂ are adjacent, they don't have any common neighbors. So, for any other contestant A_i, if A_i knows A₁, A_i doesn't know A₂, and vice versa. That means the friends of A₁ are entirely separate from the friends of A₂.Let me think about the implications of the property that any two non-adjacent contestants have exactly two common friends. So, if I pick any two contestants who don't know each other, they must have exactly two mutual friends.Now, consider a friend of A₁, say A_i. Since A_i knows A₁, and A₁ knows A₂, but A_i doesn't know A₂ (because A₁ and A₂ don't have common friends). So, A_i and A₂ are non-adjacent. According to the given property, A_i and A₂ must have exactly two common friends.One of those common friends is A₁, right? Because A₁ is connected to both A_i and A₂. So, there must be another common friend, let's say A_j, who is connected to both A_i and A₂.Similarly, for each friend A_i of A₁, there must be a corresponding friend A_j of A₂ such that A_j is connected to both A_i and A₂. This seems to create a kind of pairing between the friends of A₁ and the friends of A₂.Wait, so for every friend of A₁, there's a unique friend of A₂ that they are both connected to. That suggests that the number of friends of A₁ is equal to the number of friends of A₂ because each friend of A₁ corresponds to exactly one friend of A₂.Let me try to formalize this. Let’s denote the number of friends of A₁ as d₁ and the number of friends of A₂ as d₂. We need to show that d₁ = d₂.Since A₁ and A₂ are adjacent, and they don't share any common friends, the friends of A₁ are all non-adjacent to A₂, and vice versa. For each friend A_i of A₁, A_i and A₂ must have exactly two common friends. One of them is A₁, so there must be another friend A_j of A₂ such that A_j is connected to both A_i and A₂.This creates a bijection between the friends of A₁ and the friends of A₂. Each friend of A₁ corresponds to exactly one friend of A₂, and vice versa. Therefore, the number of friends of A₁ must equal the number of friends of A₂.So, d₁ = d₂, which means A₁ and A₂ have the same number of friends.I think that makes sense. By using the given property that non-adjacent vertices have exactly two common friends, and knowing that A₁ and A₂ don't share any friends, we can establish a one-to-one correspondence between their friends, leading to the conclusion that they have the same number of friends.**Final Answer**boxed{A_1 text{ and } A_2 text{ have the same number of friends.}}

💡Okay, so I have this problem here: there's a line given by the equation 3x + 4y + a = 0, and a circle given by x² - 2x + y² = 0. I need to find the value of 'a' such that the line is tangent to the circle. Hmm, okay, let's break this down step by step.First, I remember that for a line to be tangent to a circle, the distance from the center of the circle to the line must be equal to the radius of the circle. So, I think I need to find the center and the radius of the given circle.Looking at the circle equation: x² - 2x + y² = 0. I think I can rewrite this in the standard form by completing the square. Let me try that. So, for the x terms, I have x² - 2x. To complete the square, I take half of the coefficient of x, which is -2, so half of that is -1, and then square it, which gives me 1. So, I can write x² - 2x as (x - 1)² - 1. So, substituting back into the circle equation: (x - 1)² - 1 + y² = 0. That simplifies to (x - 1)² + y² = 1. Okay, so now it's in the standard form of a circle, which is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. So, from this, the center of the circle is at (1, 0) and the radius is √1, which is 1. Got that down.Now, moving on to the line: 3x + 4y + a = 0. I need to find 'a' such that this line is tangent to the circle. As I thought earlier, the distance from the center of the circle to the line should be equal to the radius. The formula for the distance from a point (x₀, y₀) to the line Ax + By + C = 0 is |Ax₀ + By₀ + C| / √(A² + B²). So, plugging in the values we have: the center is (1, 0), and the line is 3x + 4y + a = 0. So, A is 3, B is 4, and C is 'a'. The point (x₀, y₀) is (1, 0). Calculating the numerator: |3*1 + 4*0 + a| = |3 + a|. The denominator is √(3² + 4²) = √(9 + 16) = √25 = 5. So, the distance from the center to the line is |3 + a| / 5. And since the line is tangent to the circle, this distance must equal the radius, which is 1. So, we set up the equation: |3 + a| / 5 = 1. To solve for 'a', I can multiply both sides by 5: |3 + a| = 5. This absolute value equation means that 3 + a can be either 5 or -5. So, solving for 'a' in both cases:1. 3 + a = 5 → a = 5 - 3 → a = 22. 3 + a = -5 → a = -5 - 3 → a = -8Therefore, the possible values of 'a' are 2 and -8. Wait, let me double-check my steps to make sure I didn't make any mistakes. I completed the square correctly for the circle equation, right? Starting with x² - 2x, completing the square gives (x - 1)² - 1, so the equation becomes (x - 1)² + y² = 1. That seems correct.For the distance formula, I used the point (1, 0) and the line 3x + 4y + a = 0. Plugging into the formula: |3*1 + 4*0 + a| / √(9 + 16) = |3 + a| / 5. That looks right. Setting that equal to the radius, which is 1, gives |3 + a| = 5. Solving that gives a = 2 or a = -8. Hmm, seems solid. I don't think I made any calculation errors. Maybe I can visualize it. If a is 2, the line becomes 3x + 4y + 2 = 0. If a is -8, it becomes 3x + 4y - 8 = 0. Both of these lines should be tangent to the circle centered at (1, 0) with radius 1. I can maybe sketch a rough graph in my mind. The circle is small, radius 1, centered at (1, 0). The lines 3x + 4y + 2 = 0 and 3x + 4y - 8 = 0 should each touch the circle at exactly one point. Since the distance from the center to each line is equal to the radius, that makes sense. I think I'm confident with this solution. So, the values of 'a' are 2 and -8.

💡Okay, so I have this problem about a square ABCD with side length 2. Point E is the midpoint of side AB, and I need to find the dot product of vectors EC and ED. The options are A: √5, B: 3, C: 2√5, D: 5. Hmm, let me try to visualize the square first.Alright, square ABCD. Let me label the square in my mind: A is the bottom-left corner, B is the bottom-right, C is the top-right, and D is the top-left. So, AB is the bottom side, BC is the right side, CD is the top, and DA is the left side. E is the midpoint of AB, so it's right in the middle of the bottom side.I need to find the vectors EC and ED. Vector EC goes from point E to point C, and vector ED goes from point E to point D. Then, I have to compute their dot product.Maybe it's easier if I assign coordinates to each point. Let's place the square on a coordinate system. Let me assume point A is at (0, 0). Since the side length is 2, point B would be at (2, 0), point C at (2, 2), and point D at (0, 2). E is the midpoint of AB, so its coordinates would be halfway between A (0,0) and B (2,0). So, E is at (1, 0).Now, vector EC is from E (1, 0) to C (2, 2). To find this vector, I subtract the coordinates of E from C. So, EC = (2 - 1, 2 - 0) = (1, 2). Similarly, vector ED is from E (1, 0) to D (0, 2). So, ED = (0 - 1, 2 - 0) = (-1, 2).Now, to find the dot product of EC and ED, I can use the formula for the dot product of two vectors. If vector EC is (a, b) and vector ED is (c, d), then the dot product is a*c + b*d.So, plugging in the values from EC (1, 2) and ED (-1, 2):Dot product = (1)*(-1) + (2)*(2) = -1 + 4 = 3.Wait, that's one of the options. Option B is 3. So, is that the answer? Let me double-check my steps to make sure I didn't make a mistake.First, assigning coordinates: A(0,0), B(2,0), C(2,2), D(0,2). E is midpoint of AB, so (1,0). That seems correct.Vector EC: from E(1,0) to C(2,2). So, change in x is 2-1=1, change in y is 2-0=2. So, vector EC is (1,2). Correct.Vector ED: from E(1,0) to D(0,2). Change in x is 0-1=-1, change in y is 2-0=2. So, vector ED is (-1,2). Correct.Dot product: (1)*(-1) + (2)*(2) = -1 + 4 = 3. Yeah, that seems right.Alternatively, maybe I can solve it without coordinates, just using vector properties. Let me think.In square ABCD, all sides are equal and all angles are 90 degrees. E is the midpoint of AB, so AE = EB = 1.Vectors EC and ED can be expressed in terms of other vectors in the square. Maybe express them in terms of vectors AB and AD.Let me denote vector AB as vector b and vector AD as vector d. Since ABCD is a square, vector AB and vector AD are perpendicular and have the same magnitude, which is 2.So, vector EC can be expressed as vector EB + vector BC. Vector EB is half of AB, so vector EB = (1/2) vector AB = (1/2)b. Vector BC is equal to vector AD, which is vector d. So, vector EC = (1/2)b + d.Similarly, vector ED can be expressed as vector EA + vector AD. Vector EA is equal to -vector EB, so vector EA = -(1/2)b. Vector AD is vector d. So, vector ED = -(1/2)b + d.Now, to find the dot product EC · ED, substitute the expressions:EC · ED = [(1/2)b + d] · [-(1/2)b + d]Let me expand this:= (1/2)b · (-1/2)b + (1/2)b · d + d · (-1/2)b + d · dSimplify each term:First term: (1/2)*(-1/2)*(b · b) = (-1/4)|b|²Second term: (1/2)*(b · d)Third term: (-1/2)*(d · b)Fourth term: (d · d) = |d|²Now, since b and d are perpendicular, their dot product is zero. So, the second and third terms cancel each other out.So, EC · ED = (-1/4)|b|² + |d|²Since |b| = |d| = 2, because each side of the square is length 2.So, EC · ED = (-1/4)*(4) + 4 = (-1) + 4 = 3.Hmm, same result as before. So, that confirms it. The dot product is 3. So, the answer is B: 3.I think that's solid. I approached it both with coordinate geometry and with vector decomposition, and both gave me the same answer. So, I feel confident that 3 is the correct answer.**Final Answer**boxed{B}

💡First, I need to determine the total number of employees in the company. Let's denote the total number of employees as ( E ).60% of the employees drive to work, so the number of employees who drive to work is ( 0.6E ).The remaining 40% of the employees do not drive to work, which means ( 0.4E ) employees do not drive.Of the employees who don't drive to work, half take public transportation. Therefore, the number of employees who take public transportation is ( 0.5 times 0.4E = 0.2E ).We are given that there are 40 more employees who drive to work than take public transportation. This can be expressed as:[0.6E = 0.2E + 40]Now, let's solve for ( E ):[0.6E - 0.2E = 40][0.4E = 40][E = frac{40}{0.4}][E = 100]So, the company has (boxed{100}) employees.