Interesting Combinatorial Identities; e.g. $\sum_{k=0}^n {n\choose k}^2 = {2n\choose n}$

Question

I came across the following combinatorial identity:

$$\sum_{k=0}^n {n\choose k}^2 = {2n\choose n}$$

Here's the kind of proof which caught my interest:

$\sum_k {n \choose k}^2 = \sum_k {n \choose k}{n \choose n - k}$, and this represents the number of ways we might choose a committee of $n$ people out of a group of $2n$ people. On the other hand, ${2n \choose n}$ represents the same thing. So the result follows.

Now, I'm looking for some nice combinatorial identities which are, in spirit, similar to such an identity.

Thank you.

There's quite a lot of nice things here: https://en.wikipedia.org/wiki/Binomial_coefficient including a proof of your identity — GiantTortoise1729, Jul 05 '15 at 23:08
The Google search "combinatorial argument" OR "combinatorial proof" site:math.stackexchange.com will turn up many examples. — Brian M. Scott, Jul 06 '15 at 03:48
As an aside, letting $n=\dfrac12$ in the identity you quoted, we get a formula for $\pi$. — Lucian, Jul 06 '15 at 07:16

score 3 · Accepted Answer · answered Jul 05 '15 at 23:18

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There is a book called "Proofs that Really Count: The Art of Combinatorial Proof" by Arthur T. Benjamin and Jennifer J. Quinn, which might be of interest to you.

answered Jul 05 '15 at 23:18

Vincent Pfenninger

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Also the downloadable book "A-B" on methods for automatically discovering and proving combinatorial identities by Wilf, et al. – hardmath Jul 06 '15 at 02:38

Interesting Combinatorial Identities; e.g. $\sum_{k=0}^n {n\choose k}^2 = {2n\choose n}$

1 Answers1