004 Sample Final A, Problem 17

How many ways can a committee of four people can be selected from five married couples if no committee is to include both husband-and-wife pairs? (simplify your answer to a single number)

Foundations
What is the maximum number of people we can choose from each couple to serve on the committee?
Answer:
We can at most choose one person from each couple to serve on the committee.

Solution:

Step 1:
We can at most choose 1 person from the husband and wife pairs.
So, we have $5 \choose 4$ ways to pick a 4 people from each couple.

Step 2:
But, then for each couple, we could choose the husband or the wife to serve on the committee.
So, there are $2{5 \choose 4}=10$ ways to select a committee of four people.

Final Answer:
10

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