The number of square matrices of order 5 with entries from the set $$\{0, 1\}$$, such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1, is

We need to find the number of $$5 \times 5$$ matrices with entries from $$\{0, 1\}$$ such that each row sum and each column sum equals 1.

A matrix with entries 0 and 1 where every row and every column sums to 1 must have exactly one 1 in each row and exactly one 1 in each column. Such a matrix is called a permutation matrix.

Each permutation matrix of order $$n$$ corresponds to a unique permutation of $$\{1, 2, 3, 4, 5\}$$:
the 1 in row 1 can be in any of 5 columns,
the 1 in row 2 can be in any of the remaining 4 columns,
the 1 in row 3 can be in any of the remaining 3 columns,
and so on. This shows that the number of such matrices is $$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$$.

The answer is Option B: 120.