- #1

Eclair_de_XII

- 1,083

- 91

## Homework Statement

"Explain combinatorially (not algebraically) why ##\binom n k=\binom n {n-k}##. In other words, explain why this is true using words, and not algebraic manipulations."

## Homework Equations

##\binom n k = \frac{n!}{k!(n-k)!}##

## The Attempt at a Solution

"Suppose you have a set of ##n## elements ##S_n##. Partition this set into two disjoint subsets of ##k## elements ##S_k## and ##n-k## elements ##S_{n-k}##. There are ##\frac{n!}{(n-k)!}## total ways to create ##S_k##, and of those total ways, a given set of ##k## elements will have ##k!## permutations. So there are ##\binom n k = \frac{n!}{k!(n-k)!}## unduplicated sets of ##k## elements."

I'm having trouble seeing as how this is the same as creating all possible ##S_{n-k}##.