Assume that you have a total of 16 people on the board: 4 out-of-state seniors, 5 in-state seniors, 2 out-of-state non-seniors, and 5 in-state non-seniors.

Question

University rules require that at least one in-state student and at least one senior hold one of the three offices. Note that if individuals change offices, then a different selection exists.

In how many ways can the officers be chosen while still conforming to University rules?

Answer 1

All your other problems are basically the same. This one requires a little more thought.

The number of sets of officers that include at least one senior and one in state equals the total number of sets minus those that only include out-of-state or only include non-seniors. The way to get the number that only include out-of-state or only include non-seniors is to realize the number that include out-of-state and only non-senior (the double counted) equals the number that are out-of-state non-seniors, but the number of out-of-state non-seniors is 2, so we can't have all 3 offices with out-of-state non-seniors, so no double counting.

There are 4+2 = 6 out-of-state and 5+2 = 7 non-seniors.

Thus, the number of possible sets of officers, where P(n,x) is the number of selections of x from n where order matters, is P(16,3) - P(7, 3) - P(6,3) = 16*15*14 - 7*6*5 - 6*5*4 = 3030