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

With either 0 or 1 as an entry, a sum of 4 is possible only with four 1s and one Zero.

So, each column and row must have four 1s and one 0.

Let us consider the first row, we can put the O in any of the 5 places. 

So, number of ways of placing 1 zero in the first row=5.

In the second row, we cannot put the zero in the same column as that put by the first, as it will result in two )s in the same column. So, the second 0 can be but in any of the remaining 4 places. 

Similarly, for the third zero, we will have 3 ways to do so. And so on.

So, total number of ways to put 5 zeroes in the 5$$\times\ 5$$ matrix is 5*4*3*2*1= 120.