Grand Ping Pong Tournament

Number Theory Level 3

Two people are competing with each other during a Ping-Pong tournament, whoever wins four matches first wins (No tie game). How many possible scenarios are there? (Ex: WWLWW and LWWWW count as different scenarios)

The answer is 70.

2 solutions

Kevin Xu
Sep 16, 2019

As an upgrade version of Ping Pong Tournament the idea is similar. $\\$

The game may last from 4(One person WWWW) up to 7 (One person lose three wins four games) $\\$ So we break down the possible scenarios into 4 games, 5 games, 6 games, and 7 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: WWWWLL is not possible).

WWWW	1
_ _ _ _ W	$C^3_4=4$
_ _ _ _ _W	$C^3_5 = 10$
_ _ _ _ _ _ W	$C^3_6 = 20$

$(1 + 4 + 10 + 20) \times 2 = 70$

Richard Desper
Sep 17, 2019

Similar to logic of best-of-5 competition, there are $\binom{7}{4}$ ways for Player 1 to win and $\binom{7}{4}$ ways for Player 2 to win.

And, yes, this generalizes. If two players compete until one has won $n$ matches, there are 2* $\binom{2n-1}{n}$ possible tournaments.

Grand Ping Pong Tournament

The answer is 70.

2 solutions

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