Question

Give an algebraic proof that 2n choose 2 = 2 (n choose 2) +n^2

Answer 1

$\dbinom{2n}2=\dfrac{(2n!)}{2!(2n-2)!}=\dfrac{2n(2n-1)}2=n(2n-1)=2n^2-n$

$2\dbinom n2+n^2=2\dfrac{n!}{2!(n-2)!}+n^2=n(n-1)+n^2=2n^2-n$