HMMT 十一月 2022 · 冲刺赛 · 第 2 题

HMMT November 2022 — Guts Round — Problem 2

[5] The English alphabet, which has 26 letters, is randomly permuted. Let p be the probability that AB , 1 CD , and EF all appear as contiguous substrings. Let p be the probability that ABC and DEF both appear 2 p 1 as contiguous substrings. Compute . p 2

解析

[5] The English alphabet, which has 26 letters, is randomly permuted. Let p be the probability that 1 AB , CD , and EF all appear as contiguous substrings. Let p be the probability that ABC and DEF both 2 p 1 appear as contiguous substrings. Compute . p 2 Proposed by: Ankit Bisain, Luke Robitaille Answer: 23 Solution: There are 23! ways to arrange the alphabet such that AB , CD , and EF all appear as contiguous substrings: treat each of these pairs of letters as a single merged symbol, which leaves 23 symbols to permute. Similarly, there are 22! ways to arrange the alphabet such that ABC and DEF both appear as contiguous substrings. Thus, p = 23! / 26! and p = 22! / 26!, so the answer is 23! / 22! = 23. 1 2