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

Question

Assume that you have a total of 13 people on the board: 3 out-of-state seniors, 4 in-state seniors, 1 out-of-state non-senior, and 5 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.

Answer 1

Seems like just working it out by counting is easiest and surest way. Note I assume one in-state senior meets both criteria. So fill first place and work out what is required for other two. There are 3 OSS, 4ISS, 1 OSNS, and 5 ISNS

ISS - - is 4 x 12 x 11

OSS IS - is 3 x 9 x 11

OSS OS IS is 3 x 3 x 9

OSNS ISS - 1 x 4 x 11

OSNS OSS IS 1 x 3 x 9

OSNS ISNS S 1 x 5 x 7

ISNS S - 5 x 7 x 11

ISNS NS S 5 x 5 x 7

Add those up. Please check.