Assume that you have a total of 13 people on the committee: 4 black males, 4 black females, 1 white male, and 4 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?
Any ideas on how to solve this problem?
I would start with the number of ways of selecting three people with no restrictions, which is
13!/(3!*10!)= 286
and subtract the number of possible combinations that do not meet the requirements. For example, the number of combinations with no blacks (with only 5 nonblacks to choose among)is
5!/(3!*2!) = 10. Those combinations must be excluded. The number of ways of picking combinations with no females (with 5 males to choose among) is also 10.
That leaves 266
P(13,3)=1716 (the total amount of combinations)
P(5,3)=60 (the total amount of non-blacks)
P(5,3)=60 (the total amount of non-females)
So subtract from the total the amount of combinations that won't work. So,
1716-60-60=1596
To solve this problem, we can use the principle of inclusion-exclusion. We'll calculate the total number of ways to choose the officers and subtract the ways that violate the party rules.
First, let's calculate the total number of ways to choose the officers without any restrictions. We have 13 people and 3 offices, so the total number of ways is 13P3 (permutations of 13 people taken 3 at a time) which can be calculated as:
13P3 = 13! / (13-3)! = 13! / 10! = 13 * 12 * 11 = 1,716
Now, let's calculate the number of ways that violate the party rules. There are two scenarios to consider:
1. No black person holds an office: In this case, we have the white male and four white females to choose from for the three offices. So the number of ways this can happen is 5P3 = 5! / (5-3)! = 5! / 2! = 5 * 4 = 20.
2. No female holds an office: In this case, we have the four black males, the white male, and the four white females to choose from for the three offices. The number of ways this can happen is 9P3 = 9! / (9-3)! = 9! / 6! = 9 * 8 * 7 = 504.
Finally, we'll subtract the number of ways that violate the party rules from the total number of ways:
Total number of ways to choose officers while conforming to party rules = Total number of ways - Number of ways that violate the party rules
= 1,716 - (20 + 504)
= 1,716 - 524
= 1,192
Therefore, there are 1,192 ways to choose the officers while still conforming to the party rules.