Pickleball is a game played by two people The first player t

Pickleball is a game played by two people. The first player to score 18 points wins, provided that player is at least 2 points ahead. Otherwise, play continues until one player gets 2 points ahead, at which time he is declared the winner. However, the first player to reach a score of 24 wins the game even if he is not 2 points ahead.

Mike and Harvey are playing Pickleball and they wonder how many possible outcomes there could be. Here are some possible final scores:

Mike wins 18 to 0
Mike wins 18 to 1
Mike wins 18 to 2
Harvey wins 23 to 21
Harvey wins 24 to 22
Harvey wins 24 to 23

This is only a partial list of possible outcomes. How many outcomes would be in the list if ALL possible outcomes were listed?

Solution

Their are two set of outcome

First the outcomes can be

18 - 0

18-1

...

...

18-16

and say Mike wins.

So their are (16+1)=17 possible choices.

Now since any one can win this choices are valid for both of them so 2*17 = 34 choices

At the second case where the situation is

18-17

their lies a possibility of the following choices

24-17,24-18, ...24-23(altogether 7 choices) and Mike wins

OR

if the first stage yields with 18-18

then thhe cases are

24-18-,24-19,...24-23 (altogether 6 choices ) and Mike wins.

So total no. choices are 6+7=13 ways

Now since we have chosen arbitrarily one the actual no. of ways is 2*13 =26

So there will be altogether 34+26 = 60 kinds of final result can appear.