Hello, oldguynewstudent!

I agree with your reasoning and your answer.

A university has 120 incoming freshman that will be assigned to on-campus housing.

The only remaining dorm holds 105 students and contains 42 doubles and 7 triples.

In how many ways can the university select 105 students to house in this dorm

and then arrange those students into roommate pairs and triples?

First, choose 105 from the 120 incoming freshmen.

. . ways.

Second, partition the 105 students into 42 pairs and 7 triples.

. . ways.

Finally, the answer is the product of the two quantities above.

I'll letcrank out the answer . . . I'll wait in the car.you