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HMMT 十一月 2011 · 冲刺赛 · 第 6 题

HMMT November 2011 — Guts Round — Problem 6

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

  1. [ 6 ] To survive the coming Cambridge winter, Chim Tu doesn’t wear one T-shirt, but instead wears upto FOUR T-shirts, all in different colors. An outfit consists of three or more T-shirts, put on one ontop of the other in some order, such that two outfits are distinct if the sets of T-shirts used are differentor the sets of T-shirts used are the same but the order in which they are worn is different. Given that
    Chim Tu changes his outfit every three days, and otherwise never wears the same outfit twice, howmany days of winter can Chim Tu survive? (Needless to say, he only has four t-shirts.)
    4 ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011 — GUTS ROUNDTH
    Round 3
    4 2

英文原题

[6]
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 TH ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011— GUTS ROUND
School Team Team ID#

解析

英文解析

  1. [ 6 ] To survive the coming Cambridge winter, Chim Tu doesn’t wear one T-shirt, but instead wears upto FOUR T-shirts, all in different colors. An outfit consists of three or more T-shirts, put on one ontop of the other in some order, such that two outfits are distinct if the sets of T-shirts used are differentor the sets of T-shirts used are the same but the order in which they are worn is different. Given that
    Chim Tu changes his outfit every three days, and otherwise never wears the same outfit twice, howmany days of winter can Chim Tu survive? (Needless to say, he only has four t-shirts.)
    Guts Round
    Answer: 144 We note that there are 4 choices for Chim Tu’s innermost T-shirt, 3 choices for thenext, and 2 choices for the next. At this point, he has exactly 1 T-shirt left, and 2 choices: either heputs that one on as well or he discards it. Thus, he has a total of 4 × 3 × 2 × 2 = 48 outfits, and cansurvive for 48 × 3 = 144 days.
    4 2