Using summation((\stackrel{n}{k})xkyn-k) = (x+y)n, I let x = y = 1. This should then result in summation((\stackrel{n}{k})*1*1) = (1 + 1)n = 2n.
Expanding the summation, I get
(\stackrel{n}{0}) + (\stackrel{n}{1}) + ... +(\stackrel{n}{n}) = 2n.
Solving this results in...