Ping Pong Tournament

The game may last from 3(One person WWW) up to 5 (One person lose two wins three games) $\\$ So we break down the possible scenarios into 3 games, 4 games, and 5 games. $\\$ - We only count the ways a person can win the game then times it by two for the sake of simplicity. $\\$ - We also assume the person wins the last game, otherwise, the game would end early before reaching the total number of matches (Ex: WWWLL is not possible).

$(1 + 3 + 6) \times 2 = 20$

There are $\binom{5}{3}$ ways for Player 1 to win and $\binom{5}{3}$ ways for Player 2 to win.

Rather than divide the analysis into cases based on the number of games played, it's simpler to consider each series a Best-of-5, with a convention that, should the winner need fewer than five games to get three wins, any remaining games are treated as wins for the other player. (Thus, for example, WWW ~ WWWLL.)

So we need only add the $\binom{5}{3}$ ways to select exactly 3 'W's from 5 slots to the $\binom{5}{3}$ ways to select exactly 3 'L's from 5 slots.