Two players P1 and P2 play a series of 2n games. Each game can result in either a win or a loss for P1. The total number of ways in which P1 can win the series of these games is equal to

P1 must win atleast n+1 games. Let P1 win n+r games (r = 1, 2,..., n). Therefore, corresponding number of ways is  2nCn+r. The total number of ways is ∑r=1n 2nCn+r=2nCn+1+2nCn+2+⋯+2nC2n=22n−2nCn2