HMMT 二月 2003 · 团队赛 · 第 1 题
HMMT February 2003 — Team Round — Problem 1
题目详情
英文原题
- (a) [10] How many k -configurations are there of a set that has n elements?
(b) [10] How many k -configurations that have m elements are there of a set that hasn elements?
解析
英文解析
- (a) How many k -configurations are there of a set that has n elements?
( )
Solution: An n -element set has subsets of size k , and we can construct a k -nkconfiguration by independently choosing, for each subset, whether or not to include it,
( )nkso there are 2 k -configurations.
(b) How many k -configurations that have m elements are there of a set that has nelements?
( ) ( )
( )nnk
Solution: Again, an n -element set has subsets of size k , so there are k k mconfigurations with m elements.