HMMT 二月 2010 · 冲刺赛 · 第 12 题
HMMT February 2010 — Guts Round — Problem 12
题目详情
- [ 7 ] How many different collections of 9 letters are there? A letter can appear multiple times in acollection. Two collections are equal if each letter appears the same number of times in both collections.
13 ANNUAL HARVARD-MIT MATHEMATICS TOURNAMENT, 20 FEBRUARY 2010 — GUTS ROUNDth
英文原题
[ 7 ] How many different collections of 9 letters are there? A letter can appear multiple times in a
collection. Two collections are equal if each letter appears the same number of times in both collections.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13 th ANNUAL HARVARD-MIT MATHEMATICS TOURNAMENT, 20 FEBRUARY 2010 — GUTS ROUND
解析
英文解析
- [ 7 ] How many different collections of 9 letters are there? A letter can appear multiple times in acollection. Two collections are equal if each letter appears the same number of times in both collections.
( )
Answer: We put these collections in bijections with binary strings of length 34 containing 934
zeroes and 25 ones. Take any such string - the 9 zeroes will correspond to the 9 letters in the collection.9
If there are n ones before a zero, then that zero corresponds to the ( n + 1)st letter of the alphabet.
This scheme is an injective map from the binary strings to the collections, and it has an inverse, so the
( )
number of collections is .34
13 ANNUAL HARVARD-MIT MATHEMATICS TOURNAMENT, 20 FEBRUARY 2010 — GUTS ROUNDth