Assume that you have a total of 9 people on the committee: 2 black males, 1 black female, 1 white male, and 5 white females. Party rules require that at least one black and at least one female hold one of the three offices.
In how many ways can the officers be chosen while still conforming to party rules?
To calculate the number of ways the officers can be chosen while conforming to party rules, we need to consider several scenarios.
Scenario 1: One black male, one black female, and one other person (either white male or white female):
- There are 2 options for choosing the black male.
- There is 1 option for choosing the black female.
- There are 7 remaining people, out of which we can choose 1 person.
So, the number of ways to choose the officers in this scenario is 2 * 1 * 7 = 14.
Scenario 2: One black male, one white female, and one other person (either black female or white male):
- There are 2 options for choosing the black male.
- There are 5 options for choosing the white female.
- There are 2 remaining people (either black female or white male), out of which we can choose 1 person.
So, the number of ways to choose the officers in this scenario is 2 * 5 * 2 = 20.
Scenario 3: One white male, one black female, and one other person (either black male or white female):
- There is 1 option for choosing the white male.
- There is 1 option for choosing the black female.
- There are 7 remaining people, out of which we can choose 1 person.
So, the number of ways to choose the officers in this scenario is 1 * 1 * 7 = 7.
Therefore, the total number of ways to choose the officers while conforming to party rules is the sum of the number of ways for each scenario:
Total = Scenario 1 + Scenario 2 + Scenario 3
Total = 14 + 20 + 7
Total = 41
Hence, there are 41 ways to choose the officers while still conforming to party rules.