PUMaC 2008 · 组合（B 组） · 第 7 题

PUMaC 2008 — Combinatorics (Division B) — Problem 7

(5 points) In how many ways can Alice, Bob, Charlie, David, and Eve split 16 marbles among themselves so that no two of them have the same number of marbles?

解析

In how many ways can Alice, Bob, Charlie, David, and Eve split 18 marbles among themselves so that no two of them have the same number of marbles? 2 Combinatorics ( ANS: 2160: Suppose we have a valid 5-tuple a < b < c < d < e . Let A = a , B = b − a − 1, C = c − b − 1, D = d − c − 1 and E = e − d − 1. Now we have A, B, C, D ≥ 0 and A + B + C + D + E = 8 We can just enumerate the solutions: for 18: 10100 10011 10003 02000 01101 01020 01012 01004 00210 00202 00121 00113 00105 00040 00032 00024 00016 00008 so there are 18 ∗ 5! solutions CB: AP)