---
title: "Is there another way of expressing $\sum\limits_{k=1}^n \binom{n}{k} \cdot (n-k)! \cdot k!$?"
description: "I have stumbled across the formula $ \sum\limits_{k=1}^n \binom{n}{k} \cdot (n-k)! \cdot k!$ in some applied context. It looks as though there might be a nice combinatorial background to this..."
url: https://extraproxies.com/is-there-another-way-of-expressing-sumlimits_k1n-binomnk-cdot-n-k-cdot-k
date: 2024-01-18
modified: 2024-01-18
author: "ExtraProxies"
categories: ["Proxy Feed"]
tags: ["$\sum\limits_{k=1}^n", "another", "expressing", "there", "\binom{n}{k}", "\cdot"]
type: post
lang: en
---

# Is there another way of expressing $\sum\limits_{k=1}^n \binom{n}{k} \cdot (n-k)! \cdot k!$?

I have stumbled across the formula $ \sum\limits_{k=1}^n \binom{n}{k} \cdot (n-k)! \cdot k!$ in some applied context. It looks as though there might be a nice combinatorial background to this formula. Is there another way to express it? Maybe a term that only depends on n? I am thinking of the famous equation expressing $ 2^n$ as a sum of binomial coefficients and I am hoping that this may be a similar (albeit more complicated) instance. TIA!