# Counting Problem

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• May 9th 2009, 07:48 PM
bearej50
Counting Problem
The Columbus Crew are playing the D. C. United in a best three-out-of-five championship soccer series.
a. What are the possible outcomes of the series?
b. How many outcomes require exactly four games to determine the champion?

c. How many ways can D. C. United win the championship?
• May 10th 2009, 08:27 AM
Plato
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Quote:

Originally Posted by bearej50
The Columbus Crew are playing the D. C. United in a best three-out-of-five championship soccer series.
a. What are the possible outcomes of the series?
b. How many outcomes require exactly four games to determine the champion?

c. How many ways can D. C. United win the championship?

Suppose that $\displaystyle \mathbb{C}$ stands for “Columbus Crew wins”.
Suppose that $\displaystyle \mathbb{B}$ stands for “D. C. United wins”.
Make a ‘truth table’ type listing (five columns and thirty-two rows) of all possible outcomes of the five games.
Once that is done, it is just a matter of counting.
• May 10th 2009, 09:06 AM
Soroban
Hello, bearej50!

Quote:

The Columbus Crew are playing the D. C. United in a best 3-out-of-5 championship soccer series.

a. What are the possible outcomes of the series?

If we're asked to LIST all the outcomes, the other two questons are trivial.

. . $\displaystyle \begin{array}{ccccc}D&D&D \\ D&D&C&D \\ D&D&C&C&D \\ D&D&C&C&C \\ D&C&D&D \\ D&C&D&C&D \\ D&C&D&C&C \\ D&C&C&D&D \\ D&C&C&D&C \\ D&C&C&C\end{array}$

. . $\displaystyle \begin{array}{ccccc}C&D&D&D \\ C&D&D&C&D \\ C&D&D&C&C \\ C&D&C&D&D \\ C&D&C&D&C \\ C&D&C&C \\ C&C&D&D&D \\ C&C&D&D&C \\ C&C&D&C \\ C&C&C \end{array}$

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b. How many outcomes require exactly 4 games to determine the championship? . Six

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c. How many ways can D. C. United win the championship? . Ten