HMMT 十一月 2025 · GEN 赛 · 第 3 题

HMMT November 2025 — GEN Round — Problem 3

Ashley fills each cell of a 3 × 3 grid with some of the numbers 1, 2, 3, and 4 (possibly none or several). Compute the number of ways she can do so such that each row and each column contains each of 1, 2, 3, and 4 exactly once. (One such grid is shown below.) 1 2 3 4 4 1 2 3 3 2 1 4 2

解析

Ashley fills each cell of a 3 × 3 grid with some of the numbers 1, 2, 3, and 4 (possibly none or several). Compute the number of ways she can do so such that each row and each column contains each of 1, 2, 3, and 4 exactly once. (One such grid is shown below.) © 2025 HMMT 1 2 3 4 4 1 2 3 3 2 1 4 Proposed by: Ashley Zhu Answer: 1296 Solution: We consider the placements of each of 1, 2, 3, and 4 separately. Observe that there are 3! = 6 ways to place the 1’s, as each row must have a 1 and they must be in different columns. The 4 same holds for 2, 3, and 4. Thus, the answer is 6 = 1296 . 2