HMMT 二月 2005 · 冲刺赛 · 第 4 题

HMMT February 2005 — Guts Round — Problem 4

[6] What is the probability that in a randomly chosen arrangement of the numbers and letters in “HMMT2005,” one can read either “HMMT” or “2005” from left to right? (For example, in “5HM0M20T,” one can read “HMMT.”)

解析

What is the probability that in a randomly chosen arrangement of the numbers and letters in “HMMT2005,” one can read either “HMMT” or “2005” from left to right? (For example, in “5HM0M20T,” one can read “HMMT.”) Solution: 23 / 144 1 ( ) 8 4! To read “HMMT,” there are ways to place the letters, and ways to place the 4 2 ( ) 8 4! numbers. Similarly, there are arrangements where one can read ”2005.” The 4 2 ( ) 8 number of arrangements in which one can read both is just . The total number of 4 8! arrangements is , thus the answer is 4 ( ) ( ) ( ) ( ) 8 8 8 4! 4!