---
title: "combination"
type: concept
related: [Permutation, Combination]
source: https://www.jemoka.com/posts/kbhcombination/
confidence: high
status: active
---

A combination is a choice task which shows that order does not matter.
\begin{equation} \mqty(n \\k) = \frac{n!}{k!(n-k)!} = n! \times 1 \times \frac{1}{k!} \times \frac{1}{(n-k)!} \end{equation}
This could be shown as follows: we first permute the group of people \(n\) (\(n!\)); take the first \(k\) of them (only 1 chose); we remove the overcounted order from the \(k\) subset chosen (\(\frac{1}{k!}\)),; we remove the overcounted order from the \(n-k\) subset (\(\frac{1}{(n-k)!}\)).
There are many ways of making this happen in code:
n_choose_k = math.factorial(n) / (math.factorial(k) * math.factorial(n-k)) n_choose_k = math.comb(n,k) n_choose_k = itertools.combinations(range(n), k)
