PUMaC 2019 · 组合（A 组） · 第 4 题

PUMaC 2019 — Combinatorics (Division A) — Problem 4

Kelvin and Quinn are collecting trading cards; there are 6 distinct cards that could appear in a pack. Each pack contains exactly one card, and each card is equally likely. Kelvin buys packs until he has at least one copy of every card, then he stops buying packs. If Quinn is missing exactly one card, the probability that Kelvin has at least two copies of the card Quinn is missing is expressible as m/n for coprime positive integers m, n . Determine m + n .

解析

Kelvin and Quinn are collecting trading cards; there are 6 distinct cards that could appear in a pack. Each pack contains exactly one card, and each card is equally likely. Kelvin buys packs until he has at least one copy of every card, then he stops buying packs. If Quinn is missing exactly one card, the probability that Kelvin has at least two copies of the card Quinn is missing is expressible as m/n for coprime positive integers m, n . Determine m + n . Proposed by Sam Mathers. 1 Answer: 191 . Solution: However, we also have the probabilities for each of the new cards that appear. This 5 4 1 1 1 is · · . . . · · since we are fixing when A appears so we have two copies of , one for A 6 6 6 6 6 5 6 5! 1 and one for the last card that isn’t A , Thus, in total, the probability is · = . 6 5! · (6 − n ) 6 (6 − n )6 We now need to sum this over all possible n , giving us 5 ∑ 1 1 1 1 1 49 = (1 + + + . . . + ) = . (6 − n )6 6 2 3 6 120 n =0 49 71 Since we computed the complement, the probability we want is 1 − = . 120 120