The value of 15∑r=1r2(15Cr15Cr) is equal to:

15∑r=1r2(15Cr15Cr)→(nCrnCr)=n−r+1r
15∑r=1r2(15Cr15Cr)=15∑r=1r2(n−r+1r)∑15r=1(nr−r2+r)=∑15r=1((n+1)r−r2)∑15r=1(nr−r2+r)=n(n+1)22−n(n+1)(2n+1)6
Put n=15, we get 15∑r=1r2(15Cr15Cr)=680