HMMT 二月 2005 · TEAM2 赛 · 第 1 题

英文解析

1. [15] Let 0 < m ≤ n be integers. How many different (i.e., noncongruent) dominoescan be formed by choosing two squares of an m × n array?
   Solution: We must have 0 ≤ a < m , 0 ≤ b < n , a ≤ b , and a and b not both 0. Thenumber of pairs ( a, b ) with b < a < m is m ( m − 1) / 2, so the answer ism ( m − 1) m − m + 22
   mn − − 1 = mn − .
   2 2