# Combinatorial proof of summation of $\sum_{k = 0}^n {n \choose k}^2= {2n \choose n}$

Can you guys help me prove this? There is a way of proving this logically but I was hoping to find a more "mathematical" proof, if possible.

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

Logical Proof:

$${n \choose k}^2 = {n \choose k}{ n \choose n-k}$$

Hence summation can be expressed as:

$$\binom{n}{0}\binom{n}{n} + \binom{n}{1}\binom{n}{n-1} + \cdots + \binom{n}{n}\binom{n}{0}$$

One can think of it as choosing $n$ people from a group of $2n$ (imagine dividing a group of $2n$ into $2$ groups of $n$ people each. I can get $k$ people from group $1$ and another $n-k$ people from group $2$. We do this from $k = 0$ to $n$)

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Just FYI, what you call a "logical proof" is known as a "combinatorial proof", and such a proof is perfectly valid and often very insightful. What I suspect you mean by "mathematical proof" is one dealing with the numerical structure of sums and combinations, which would be better called an "analytical proof". Both styles of proof are equally mathematical. – Austin Mohr May 23 '12 at 4:57
This is secretly subsumed by this question – mixedmath May 23 '12 at 5:00
You could obtain the same combinatorial proof by noting that $\binom{2n}{n}$ counts the number of paths from $(0,0)$ to $(n,n)$ on an $n\times n$ grid. – Holdsworth88 May 23 '12 at 6:03
I think your combinatorial proof is really nice, and you should not be unhappy with it. – MJD May 23 '12 at 13:54
Incidentally, this is a special case of math.stackexchange.com/questions/337923/…. – Timere Nov 16 '13 at 10:26

The combinatorial explanation is straightforward. There's also a roundabout approach through what are called "generating functions." The binomial theorem tells us that

$$(1+x)^n(x+1)^n=\left(\sum_{a=0}^n\binom{n}{a}x^a\right)\left(\sum_{b=0}^n\binom{n}{b}x^{n-b}\right)=\sum_{c=0}^{2n}\left(\sum_{a+n-b=c}\binom{n}{a}\binom{n}{b}\right)x^c$$

The $x^n$ coefficient of the above occurs with $c=n$, wherein the coefficient is

$$\sum_{a+n-b=n}\binom{n}{a}\binom{n}{b}=\sum_{a=0}^n\binom{n}{a}^2.$$

However, the $x^n$ coefficient of $(1+x)^n(x+1)^n=(1+x)^{2n}$, again by the binomial theorem, is

$$\binom{2n}{n}.$$

Equating the two gives the result.

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+1. @Lance C Do you see that this is the same as your "logical" proof? Choosing $n$ people from $2n$ is what you are doing when you look at the coefficient of $x^n$ in the expansion. – user17762 May 23 '12 at 5:12
Can you explain me in more detail... why $\sum_{a=0}^n\binom{n}{a}x^a$ * $\sum_{b=0}^n\binom{n}{b}x^{n-b}$ is equal to $\sum_{c=0}^{2n} \sum_{a+n-b=c}\binom{n}{a}\binom{n}{b} x^c$ please – Salvattore Jul 25 '14 at 4:59
@Salvattore $\sum_{a,b}\binom{n}{a}\binom{n}{b}x^{a+n-b}$, then collect all the coefficients of $x^c$ for fixed $c$ into $\sum_{a+n-b=c}\binom{n}{a}\binom{n}{b}$ to get $\sum_c\left[\sum_{a+n-b=c}\binom{n}{a}\binom{n}{b}\right]x^c$. If you want to work with generating functions, or even just plain polynomials, you need to be able to collect like terms, it is virtually a requirement. – anon Jul 29 '14 at 17:23

$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,{\rm e}^{#1}\,} \newcommand{\fermi}{\,{\rm f}} \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{{\rm i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}} \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,} \newcommand{\sech}{\,{\rm sech}} \newcommand{\sgn}{\,{\rm sgn}} \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$ \begin{align} \color{#66f}{\large\sum_{k\ =\ 0}^{n}{n \choose k}^{2}}&= \sum_{k\ =\ 0}^{n}{n \choose k} \oint_{\verts{z}\ =\ 1}{\pars{1 + z}^{n} \over z^{k + 1}}\,{\dd z \over 2\pi\ic} =\oint_{\verts{z}\ =\ 1}{\pars{1 + z}^{n} \over z} \sum_{k\ =\ 0}^{n}{n \choose k}\pars{1 \over z}^{k}\,{\dd z \over 2\pi\ic} \\[5mm]&=\oint_{\verts{z}\ =\ 1}{\pars{1 + z}^{n} \over z} \pars{1 + {1 \over z}}^{n}\,{\dd z \over 2\pi\ic} =\oint_{\verts{z}\ =\ 1}{\pars{1 + z}^{2n} \over z^{n + 1}}\,{\dd z \over 2\pi\ic} =\color{#66f}{\large{2n \choose n}} \end{align}

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Since $\dbinom n k= \dbinom n {n-k}$, the identity $$\sum_{k=0}^n \binom n k ^2 = \binom {2n} n$$ is the same as $$\sum_{k=0}^n \binom n k \binom n {n-k} = \binom {2n} n.$$ So say a committee consists of $n$ Democrats and $n$ Republicans, and one will choose a subcommittee of $n$ members. One may choose $k$ Democrats and $n-k$ Republicans in $\dbinom n k \cdot \dbinom n {n-k}$ ways. The number of Democrats is in the set $\{0,1,2,\ldots,n\}$, thus ranging from all Republicans to all Democrats. The sum then gives the total number of ways to choose $n$ out of $2n$.

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Consider the graph underlying Pascal's triangle:

In this graph, the number at each node is a binomial coefficient and can also be thought of as the number of downward paths from the apex to that node.

The left side of the identity is the number of paths that start at the apex, go down to the $n$th row and return to the apex (let's call them round trips to $n$). By reflecting the return path about the $n$th row, we get a bijective correspondence between return trips to $n$ and paths from the apex to the central node of the $2n$th row. This is nothing but the central binomial coefficient $\binom{2n}n$.

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