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Prove that if $n\geq1$ then $$\binom{2n}{n}=\sum_{k=0}^{n}(\binom{n}{k})^2$$

This is what I have so far:

By the Binomial Theorem: $$\binom{2n}{n}=\frac{(2n)!}{(2n-2)!n!}=\frac{2n!~}{n^2(n-1)^2(n-2)^2...1}$$

By the definition of summation: $$\sum_{k=0}^{n}(\binom{n}{k})^2=(\binom{n}{0})^2+(\binom{n}{1})^2+(\binom{n}{2})^2+(\binom{n}{3})^2+...+(\binom{n}{n})^2$$

By the Binomial Theorem => $$\sum_{k=0}^{n}(\binom{n}{k})^2=1^2+(\frac{n!}{(n-1)!})^2+(\frac{n!}{(n-2)!2!})^2+(\frac{n!}{(n-3)!3!})^2+...+1^2$$

$$=1+\frac{n!n!}{((n-1)!)^2}+\frac{n!n!}{((n-2)!)^2(2!)^2}+\frac{n!n!}{((n-3)!)^2(3!)^2}+...+1$$

This is where I get stuck. I'm not sure how to continue from here. I know that what I am getting is starting to look a lot like what I want it to look like but I don't know how to finish, nor am I sure that I've done it correctly to this point. Help?

Kristin
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1 Answers1

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Although this question has been answered, I have another proof.

Let $N=\{1,\ldots,n\}$, $N_2=\{1,2\}\times N$, $A$ be the set of subsets of $N$, $B=\{(X,Y)\in A\times A:|X|=|Y|\}$ and $C=\{X\subset N_2:|X|=n\}$.

It's easy to show that $$|B|=\sum_{k=0}^n\binom nk^2$$ and $$|C|=\binom {2n}n$$ Therefore, our goal is to show that $|B|=|C|$, so let's build a bijective function from $C$ to $B$.

Let be $X\subset N_2$. $X$ has $k$ elements whose first component is $1$ and $n-k$ elements whose first component is $2$. Define $$f(X)=\Big(\{v:(1,v)\in X\},\{v:(2,v)\notin X\}\Big)$$ Since $f$ is bijective, we are finished.

To say it in plain words, we can render $N_2$ as a matrix like this: $$\begin{pmatrix}1&2&\cdots& n\\1&2&\cdots& n\end{pmatrix}$$ Choosing a set $X\subset C$ is choosing $n$ terms of the matrix. This leaves $k$ terms selected in the first row and $k$ terms unselected in the second row. This is like selecting two subsets of the sime size from $N$, which is like selecting an element from $B$.

ajotatxe
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