# Number of $1$s in a binary grid

Consider a binary grid of size $4\times 4$, each of cell can either have $0$ or $1$. Among all possible $2^{16}$ arrangement how many arrangement of such grid exist in which each row and column contains even number of $1$s.

Solution which I thought

There will be $2$ possibilities for the answer of this question $1$st all ones in the $4\times 4$ grid that will count up to $1$ possible arrangement and $2$nd possibility will be $2$ ones in each row and column so how can i find the possible arrangement which will have $2$ ones in each row and column.

Am I right?

• Why is it not possible that some rows contain four 1s and some rows contain two 1s?
– TMM
Commented Jul 6, 2012 at 21:38

Let the positions be $a_{i,j}$, where $1\le i,j\le 4$. You can fill the $3\times 3$ square in the upper lefthand corner, i.e., positions $a_{i,j}$ with $1\le i,j\le 3$, any way you like. Once those $9$ positions are filled, there is exactly one way to fill the remaining $7$ positions to get an even number of $1$’s in each row and column. Can you see why? (HINT: Fill $a_{4,4}$ last.)