How do I solve this word problem?

Question

Luis and Al are competing in a table tennis match. The first person to win either 3 consecutive games or a total of 4 games wins the match. In how many different ways can their match be played?

Answer 1

I tried this approach:

Luis wins the match if he wins the games as follows:

1,2,3

2,3,4

3,4,5

We don't consider games 4,5,6 because if the first game Luis wins is game 4, it means he lost the first 3; he's out.

If Luis doesn't win 3 consecutive games, he needs to win 4 games as follows:

1,2, 4,5   (lost 1 game)

1,2, 4, 6  (lost 2 games)

1,2, 4, 7  (lost 3 games)

1,2, 5,6   (lost 2 games)

1,2, 5, 7  (lost 3 games)

We don't consider 1,2, 6,7 because if he doesn't win games 3,4,5 he'd lost these 3 consecutive games; he's out.

Also, Luis needs to win these games:

1, 3,4,5    (lost 1 game)

1, 3, 5,6   (lost 2 games)

1, 3, 5, 7  (lost 3 games)

1, 3, 6,7   (lost 3 games)

1, 4,5,6    (lost 2 games)

1, 4, 6,7   (lost 3 games)

2,3, 5,6    (lost 2 games)

2,3, 6,7    (lost 3 games)

2, 4,5,6    (lost 2 games)

2, 5,6,7    (lost 3 games)

3,4, 6,7    (lost 3 games)

3, 5,6,7    (lost 3 games)

These are all the combinations I found: 20 different ways in which Luis can win the match. Am I missing any?

 

 

Answer 2

There are 20 DIFFERENT possible ways for the outcome of 3 consecutive wins or 4 wins total.  The easiest way to explain/show why this is true is to make a list. L indicates a game for Luis, A indicates a game for Al.  According to the rules, three in a row and they win, or four total and they win.

 

Three consecutive wins. 2 outcomes:

L-L-L (Luis)

A-A-A (Al)

 

 

One wins one, the other wins all four. 4 outcomes:

A-A-L-A-A (Al won four games)

L-L-A-L-L (Luis won four games)

A-A-A-L-A (Al won four games)

L-L-L-A-L (Luis won four games)

 

One would argue the following do not count as different because they are three consecutive wins:

A-A-A-A-L

L-L-L-L-A

L-A-A-A-A

A-L-L-L-L

A-L-A-A-A

L-A-L-L-L

 

Alternate winning games. 2 outcomes:

L-A-L-A-L-A-L (Luis won four games)

A-L-A-L-A-L-A (Al won four games)

 

Alternate with two consecutive wins (three means the other one wins) in the sequence. 8 outcomes:

A-A-L-A-L-A (Al won four games)

L-A-A-L-A-L-A (Al won four games)

L-A-L-A-A-L-A (Al won four games)

L-A-L-A-L-A-A (Al won four games)

 

Transpose L and A for Luis’ wins

L-L-A-L-A-L (Luis won four games)

A-L-L-A-L-A-L (Luis won four games)

A-L-A-L-L-A-L (Luis won four games)

A-L-A-L-A-L-L (Luis won four games)

 

TWO sets of consecutive wins. 4 outcomes:

A-A-L-A-A (Al wins)

L-A-A-L-A-A (Al wins)

L-L-A-L-L (Luis wins)

A-L-L-A-L-L (Luis wins)

 

There are no more ways to combined two consecutive wins without going over four games total, and there are no more combinations that do not involve three consecutive wins.

 

Enjoy your story problems!

-Kirstina B.