Combinations in Finite

Question

Assume that you have a total of 13 people on the board: 5 out-of-state seniors, 5 in-state seniors, 1 out-of-state non-senior, and 2 in-state non-seniors. 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

I think this is the fastest way (I verified the count by counting in a program, and also verified with a slightly more complicated process).

There are 13*12*11 = 1716 ways to select.

As there are 6 out of state and 3 non-seniors, there are

6*5*4 = 120 ways to only select out-of-state and 3*2*1 = 6 ways to only select non-seniors.

There is no double counting, as you can't have all be out-of-state non-seniors (there is only one)

1716 - 120 - 6 = 1590