JEE Main · 2023 · 24 Jan 2023 · Shift 1 · Permutations and Combinations

Each row holds exactly one $1$, and so does each column, so the positions of the $1$'s set up a one-to-one matching of rows to columns. Such matrices are precisely the permutation matrices of order $5$. The $1$ in the first row can occupy any of $5$ columns, the next row any of the remaining $4$, then $3$, $2$, $1$. The number of such matrices is $5!=120$.