HMMT 十一月 2018 · 冲刺赛 · 第 19 题

HMMT November 2018 — Guts Round — Problem 19

[ 11 ] Let A be the number of unordered pairs of ordered pairs of integers between 1 and 6 inclusive, and let B be the number of ordered pairs of unordered pairs of integers between 1 and 6 inclusive. (Repetitions are allowed in both ordered and unordered pairs.) Find A − B .

解析

[ 11 ] Let A be the number of unordered pairs of ordered pairs of integers between 1 and 6 inclusive, and let B be the number of ordered pairs of unordered pairs of integers between 1 and 6 inclusive. (Repetitions are allowed in both ordered and unordered pairs.) Find A B . Proposed by: Yuan Yao Answer: 225 There are 6 · 6 ordered pairs of integers between 1 and 6 inclusive and 21 unordered pairs of integers 6 36 ( = 15 di ↵ erent pairs and 6 doubles). Then, A = + 36 = 666 and B = 21 · 21 = 441. Therefore 2 2 A B = 225. n ( n +1) 2 For general n , there are n ordered pairs of integers and unordered pairs of integers. Then 2 2 2 2 2 n ( n +1) n ( n +1) A = and B = so 2 4 ✓ ◆ 2 2 2 2 n (2( n + 1) ( n + 1) ) n ( n 1) A B = = . 4 2