Odd row and column sums in a 3x3 grid

The numbers 1 through 9 are placed in the nine cells of a $3\times3$ grid in a uniformly random arrangement. What is the probability that all three row sums and all three column sums are odd?

$1/14$. Mark odd numbers (five of them) as 1s. Each line needs an odd count of 1s, so with five 1s the row counts are $3,1,1$ and so are the column counts: the 1s fill one full row plus one full column. That gives $3\cdot3=9$ parity patterns out of $\binom95=126$.