# Entrance exams

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How many 5×5 matrices are there such that each entry is 0 or 1 and each row sum and each column sum is 4?

a)64   b)32   c)120   d)96

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We have to find all possible 5*5 matrices such that each row sum and each column sum is 4.

Given that A=((aij)) is either 0 or 1 for all i,j.

Let us take into account a possibility of such a matrix, A=

0  1  1  1  1

1  0  1  1  1

1  1  1  0  1

1  1  0  1  1

1  1  1  1  0

The above is a matrix whose entries are either 0 or 1 and also each row sum and column sum is 4.

One crucial point is to be observed, i.e., we have to place or put four 1’s in each column and row to make the column and row sum 4, and hence only one slot or place is left for a 0 to be placed in each row and column.

Therefore the number of possible places or ways we can put a 0 in a row and a column= no.of all possible 5*5 above such matrices= 5!=120………c) is the correct option.