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HMMT 十一月 2013 · 冲刺赛 · 第 13 题

HMMT November 2013 — Guts Round — Problem 13

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 9 ] Let S = { 1 , 2 , . . . , 2013 } . Find the number of ordered triples ( A, B, C ) of subsets of S such that
    A ⊆ B and A ∪ B ∪ C = S .
    2 2
解析

英文解析

  1. [ 9 ] Let S = { 1 , 2 , . . . , 2013 } . Find the number of ordered triples ( A, B, C ) of subsets of S such that
    A ⊆ B and A ∪ B ∪ C = S .
    2013 671
    Answer: 5 OR 125 Let n = 2013. Each of the n elements can be independently placed in 5
    3 1
    spots: there are 2 − 1 choices with element x in at least one set, and we subtract the 2 choices withelement x in set A but not B . Specifying where the elements go uniquely determines A, B, C , so theren 2013
    are 5 = 5 ordered triples.
    2 2