prove without using stirlings formula Solution Note the binomial expansion: (1+x)2n = (2n choose k) xk where k ranges from 0 to 2n. Plug x = 1 to see that 4n = 22n = (2n choose k) where the sum is over all k from 0 to 2n. Since each binomial coefficient is non-negative, we have 4n = (2n choose k) > (2n choose n) since this is merely one of the terms in the summation..