HMMT 十一月 2017 · 冲刺赛 · 第 4 题

HMMT November 2017 — Guts Round — Problem 4

[ 3 ] Alec wishes to construct a string of 6 letters using the letters A, C, G, and N, such that: • The first three letters are pairwise distinct, and so are the last three letters; • The first, second, fourth, and fifth letters are pairwise distinct. In how many ways can he construct the string?

解析

[ 3 ] Alec wishes to construct a string of 6 letters using the letters A, C, G, and N, such that: • The first three letters are pairwise distinct, and so are the last three letters; • The first, second, fourth, and fifth letters are pairwise distinct. In how many ways can he construct the string? Proposed by: Yuan Yao There are 4! = 24 ways to decide the first, second, fourth, and fifth letters because these letters can be selected sequentially without replacement from the four possible letters. Once these four letters are selected, there are 2 ways to select the third letter because two distinct letters have already been selected for the first and second letters, leaving two possibilities. The same analysis applies to the sixth 2 letter. Thus, there are 24 · 2 = 96 total ways to construct the string.