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Question

The number 1, 2, ...., n2 are arranged in an n×n array in the following way

1
2
3
....
n
n+1
n+2
n+3
....
2n
...



...
n2n+1
n2n+2
n2n+3
....
n2
Pick n numbers from the array such that any two numbers are in different rows and different columns. Find the sum of these numbers

A
n(n+1)3
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B
n(n+1)2
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C
n(n21)2
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D
n(n2+1)2
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Solution

The correct option is B n(n+1)2
If we denote by aij the number in the ith row and jth 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 permulation 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

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