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Question

The numbers 1, 2, ... , n2 are arranged in an n×n array in the following way: Pick n numbers from the array such that any two numbers are in different rows and different columns. then the sum of these numbers are nk=1akjk=n(n2+1)2.
If true then enter 1 and if false then enter 0
123...n
n+1n+2n+3...2n
n2n+1n2n+2n2n+3...n2

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Solution

If we denote by aij the number in the i th row and j th column then aij=(i1)n+j for all i, j = 1, 2, ... , n. Because any two numbers are in different rows and different columns, it follows that from each row and each column exactly one number is chosen. Let a1j1,a2j2,...,anjn be the chosen numbers, where j1,j2,...,jn is a permutation of indices 1, 2, ... , n. We have nk=1akjk=nk=1((k1)n+jk)=nnk=1(k1)+nk=1jk. But nk=1jk=n(n+1)2
since j1,j2,...,jn is a permutation of indices 1, 2, ... , n. It follows that nk=1akjk=n(n2+1)2.

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