PUMaC 2022 · 组合（B 组） · 第 1 题

PUMaC 2022 — Combinatorics (Division B) — Problem 1

Betty has a 4-by-4 square box of chocolates. Every time Betty eats a chocolate, she picks one from a row with the greatest number of remaining chocolates. In how many ways can Betty eat 5 chocolates from her box, where order matters?

解析

Betty has a 4-by-4 square box of chocolates. Every time Betty eats a chocolate, she picks one from a row with the greatest number of remaining chocolates. In how many ways can Betty eat 5 chocolates from her box, where order matters? Proposed by Adam Huang Answer: 73728 There are 16 · 12 · 8 · 4 ways to pick the first 4 chocolates since each chocolate has one less row that it can be picked from. Then, after 4 chocolates are picked, all rows again have an equal number of chocolates, so there are 12 ways to pick the last chocolate. Then, our answer is 16 · 12 · 8 · 4 · 12 = 73728.