Prove identity based on binomial theorem

Question

$\displaystyle \sum_{r=0}^{n-1} {2n-1 \choose r} = 2^{2n-2} $

Perhaps it can be proved by using sum of all combinations from r=0 to r=n is 2 to the power of n.

Answer 1

Hint: $$ 2\sum_{r=0}^{n-1} \binom{2n-1}{r} = \sum_{r=0}^{n-1} \left[\binom{2n-1}{r} + \binom{2n-1}{2n-1-r}\right]. $$