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