Assume that you have

a total of 15 people
on the committee: 5 black males, 5 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?

To find the number of ways the officers can be chosen while conforming to the party rules, we can use the principle of counting.

First, let's consider the three offices: President, Vice President, and Secretary.

There are three cases to consider based on the ethnicity and gender requirements:

Case 1: If one black male, one black female, and one person from any other category are selected for the three offices:

For the President, there are 5 options among the black males, and for the Vice President, there are 5 options among the black females.
Lastly, for the Secretary, there are 5 possible options among the remaining individuals (white males and white females).

So, for this case, the number of ways to choose the officers is 5 * 5 * 5 = 125.

Case 2: If one black male, two black females, and no white individuals are selected for the three offices:

For the President, there are 5 options among the black males, and for the Vice President, there are 5 options among the black females.
Since there needs to be at least one black and one female in the offices, there is only 1 option left for the Secretary, which must be a black female.

So, for this case, the number of ways to choose the officers is 5 * 5 * 1 = 25.

Case 3: If two black males, one black female, and no white individuals are selected for the three offices:

For the President, there are 5 options among the black males, and for the Vice President, there are 4 options among the remaining black males.
Since there needs to be at least one black and one female in the offices, there is only 1 option left for the Secretary, which must be a black female.

So, for this case, the number of ways to choose the officers is 5 * 4 * 1 = 20.

Now, summing up the number of ways for each case, we have:

Total number of ways = Number of ways for Case 1 + Number of ways for Case 2 + Number of ways for Case 3
= 125 + 25 + 20
= 170

Therefore, there are 170 ways the officers can be chosen while conforming to the party rules.