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HMMT 二月 2022 · COMB 赛 · 第 1 题

HMMT February 2022 — COMB Round — Problem 1

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Sets A , B , and C satisfy | A | = 92 , | B | = 35 , | C | = 63 , | A ∩ B | = 16 , | A ∩ C | = 51 , | B ∩ C | = 19. Compute the number of possible values of | A ∩ B ∩ C | .
解析
  1. Sets A , B , and C satisfy | A | = 92 , | B | = 35 , | C | = 63 , | A ∩ B | = 16 , | A ∩ C | = 51 , | B ∩ C | = 19. Compute the number of possible values of | A ∩ B ∩ C | . Proposed by: Daniel Zhu Answer: 10 Solution: Suppose | A ∩ B ∩ C | = n . Then there are 16 − n elements in A and B but not C , 51 − n in A and C but not B , and 19 − n in B and C but not A . Furthermore, there are 25 + n elements that are only in A , n only in B , and n − 7 that are only in C . Therefore, 7 ≤ n ≤ 16, so there are 10 possible values.