HMMT 十一月 2013 · 冲刺赛 · 第 13 题

HMMT November 2013 — Guts Round — Problem 13

[ 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

[ 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 with element x in set A but not B . Specifying where the elements go uniquely determines A, B, C , so there n 2013 are 5 = 5 ordered triples. 2 2