HMMT 二月 2006 · COMB 赛 · 第 1 题
HMMT February 2006 — COMB Round — Problem 1
题目详情
英文原题
- Vernonia High School has 85 seniors, each of whom plays on at least one of the school’s three varsitysports teams: football, baseball, and lacrosse. It so happens that 74 are on the football team; 26 areon the baseball team; 17 are on both the football and lacrosse teams; 18 are on both the baseball andfootball teams; and 13 are on both the baseball and lacrosse teams. Compute the number of seniorsplaying all three sports, given that twice this number are members of the lacrosse team.
解析
英文解析
- Vernonia High School has 85 seniors, each of whom plays on at least one of the school’sthree varsity sports teams: football, baseball, and lacrosse. It so happens that 74 areon the football team; 26 are on the baseball team; 17 are on both the football andlacrosse teams; 18 are on both the baseball and football teams; and 13 are on both thebaseball and lacrosse teams. Compute the number of seniors playing all three sports,
given that twice this number are members of the lacrosse team.
Answer: 11
Solution: Suppose that n seniors play all three sports and that 2 n are on thelacrosse team. Then, by the principle of inclusion-exclusion, 85 = (74 + 26 + 2 n ) −
(17 + 18 + 13) + ( n ) = 100 + 2 n − 48 + n = 52 + 3 n . It is easily seen that n = 11 .