HMMT 二月 2002 · 冲刺赛 · 第 1 题
HMMT February 2002 — Guts Round — Problem 1
题目详情
- [4] An ( l, a )- design of a set is a collection of subsets of that set such that each subset contains exactly l elements and that no two of the subsets share more than a elements. How many (2,1)-designs are there of a set containing 8 elements?
解析
- An ( l, a )- design of a set is a collection of subsets of that set such that each subset contains exactly l elements and that no two of the subsets share more than a elements. How many (2,1)-designs are there of a set containing 8 elements? ( ) 8 Solution: There are = 28 2-element subsets. Any two distinct such subsets have at 2 most 1 common element; hence, for each subset, we can decide independently whether or 28 not it belongs to the design, and we thus obtain 2 designs.