Even row and column sums in a 3 by 3 grid

The numbers $1,2,\dots,9$ are placed in a random arrangement in the nine cells of a $3\times3$ grid, one number per cell. Find the probability that all three row sums and all three column sums are even.

$0$. A row sum is even only when the row contains an even number of odd entries ($0$ or $2$). Then the total count of odd numbers across the three rows would be even, but $1,\dots,9$ contains five odd numbers, so no arrangement works.