HMMT 二月 2010 · 冲刺赛 · 第 5 题

HMMT February 2010 — Guts Round — Problem 5

[ 5 ] You have a length of string and 7 beads in the 7 colors of the rainbow. You place the beads on the string as follows – you randomly pick a bead that you haven’t used yet, then randomly add it to either the left end or the right end of the string. What is the probability that, at the end, the colors of the beads are the colors of the rainbow in order? (The string cannot be flipped, so the red bead must appear on the left side and the violet bead on the right side.)

解析

[ 5 ] You have a length of string and 7 beads in the 7 colors of the rainbow. You place the beads on the string as follows – you randomly pick a bead that you haven’t used yet, then randomly add it to either the left end or the right end of the string. What is the probability that, at the end, the colors of the beads are the colors of the rainbow in order? (The string cannot be flipped, so the red bead must appear on the left side and the violet bead on the right side.) 1 Answer: The threading method does not depend on the colors of the beads, so at the end all 5040 configurations are equally likely. Since there are 7! = 5040 configurations in total, the probability of 1 any particular configuration is . 5040