TITLE: How to get ${n \choose 0}^2+{n \choose 1}^2+{n \choose 2}^2+\cdots+{n \choose n}^2 = {x \choose y}$
QUESTION [12 upvotes]: I found this in my test book, any hints? Given $${n \choose 0}^2+{n \choose 1}^2+{n \choose 2}^2+\cdots+{n \choose n}^2 = {x \choose y}$$ Then find the value of x and y in n. According to the answer provided on last pages of that book, it's $2n \choose n$. What i dont understand is how to get the answer (there's no explanation written there). All i said is thank you so much for every activity to me and i'm sorry for  my bad english (english is not my native language) and my messy post.

REPLY [16 votes]: Let $E=\{a_1,\ldots a_n,b_1,\ldots,b_n\}$ a set with $2n$ elements. There's $2n\choose n$ subsets of $E$ with $n$ elements: $k$ elements from $\{a_1,\ldots,a_n\}$ and $n-k$ elements from $\{b_1,\ldots,b_n\}$ for some $k=0,\ldots,n$ hence we have
$${2n\choose n}=\sum_{k=0}^n{n\choose k}{n\choose n-k}=\sum_{k=0}^n{n\choose k}^2$$