Proof of combinatorial set

Question

How to give a combinatorial proof from

$$\sum_{i=1}^n {n\choose i}^2={2n\choose n}$$

I have tried to give an argument with 2n set elements colored red but i got stuck on this Thanks in advance

Answer 1

Consider the number of ways to colour $n$ elements from $2n$ red and the others blue. This is clearly the RHS.

Now split the $2n$ into two equally sized groups. In the first group, choose $i$ elements to colour red and make the rest blue. In the second group, choose $i$ elements to colour blue and make the rest red. Summing over all $i$ gives the number of ways to colour $n$ objects red and we are done.