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In how many ways can we fill a $3\times 3$ grid with $0$s and/or $1$s, so that every row and every column has an odd total? As an example, this is one allowed filling: $$\begin{array}{c | c | c} 0 & 1 & 0 \\ \hline 0 & 1 & 0 \\ \hline 1 & 1 & 1 \end{array}$$

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I will refer to the entry in the $i$th row and $j$th column as the $(i,j)$ entry.

You can fill the $(1,1)$, $(1,2)$, $(2,1)$, and $(2,2)$ entries however you'd like. Once those are fixed, then $(1,3)$, $(2,3)$, $(3,1)$, and $(3,2)$ are determined because the first two rows and the first two columns must sum to an odd number.

Finally, we need to check that there is a valid way to fill in the $(3,3)$ entry. You should convince yourself that if the sum of the entries in $(1,3)$ and $(2,3)$ is odd, then so is the sum of the entries in $(3,1)$ and $(3,2)$; similarly if one sum is even, then the other is even too. Thus, the entry in $(3,3)$ is also determined.

So, the only freedom you have is determining the entries in the upper-left four boxes: $2^4$.

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