# Number 1s in a binary grid

Consider a binary grid of size 4*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 1s.

Solution which I thought

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

Am I right?

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Why is it not possible that some rows contain four 1s and some rows contain two 1s? – TMM Jul 6 '12 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.)