JEE Main · 2025 · 02 Apr 2025 · Shift 2 · Permutations and Combinations

The eight boxes lie in three rows of sizes $3$, $3$ and $2$. The number of ways to place the five distinct letters in distinct boxes is ${}^8{P}_5=6720$. To forbid any empty row, remove by inclusion-exclusion the placements that avoid a row entirely: avoiding the row of two leaves ${}^6{P}_5=720$ ways, and avoiding either row of three leaves ${}^5{P}_5=120$ ways each, a total of $720+120+120=960$. Placements avoiding two rows at once are impossible, since at most four boxes would remain for five letters. Hence the number of valid placements is $6720-960=5760$.