Worked solution
Choose a committee subject to a married-couple restriction
A committee of 3 people is to be chosen from 4 married couples. Find the number of ways in which this committee can be chosen if
(i) all are equally eligible,
(ii) the committee must consist of 1 woman and 2 men,
(iii) all are equally eligible except that any husband and wife cannot both serve in the committee.
CJC · Lecture · Example 9
(i) Choose any three people
The four couples contain 8 distinct people. The number of committees is .
(ii) Select the required numbers of women and men
Choose 1 of the 4 women and 2 of the 4 men: .
(iii) Method 1: choose three different couples, then one spouse from each
Choose 3 of the 4 couples, then choose husband or wife from each selected couple. The count is . Selecting one spouse per selected couple ensures no married pair serves together.
(iii) Method 2: subtract committees containing a complete couple
From all 56 committees, subtract those formed by choosing 1 of 4 complete couples and 1 of the other 6 people: . A3-person committee cannot contain two complete couples, so every invalid committee is counted exactly once.
Common mistakes
Part (ii) imposes a gender composition, not a spouse exclusion. In(iii), explain why selecting a full couple does not count an invalid three-person committee more than once.