HMMT 二月 2000 · POW 赛 · 第 6 题
HMMT February 2000 — POW Round — Problem 6
题目详情
- Find the n um b er of om binations of length k that use elemen ts from a set of n distin t elemen ts, allo wing rep etition.
解析
- Sin e w e w an t to ho ose k times from n distin t elemen ts, this is equiv alen t to ho os- ing where to put n 1 "dividers" that separate the hoi es. F or example, if w e w an ted to ho ose 3 s o ops of i e ream from the a v ors ho olate, v anilla, stra wb erry , and o ee, w e an represen t the hoi e of 2 v anilla and 1 o ee b y (divider), hoi e, hoi e,(divider),(divider), ho Noti e that the hoi es of p ositions for the dividers ompletely determines whi h ele- men ts w e ho ose. Therefore, w e ha v e n + k 1 spa es to ll with n 1 dividers, so the n + k 1 n + k 1 n um b er of w a ys of doing this is or n 1 k