In how many ways can a student select $4$ out of $20$ recommended books if exactly $3$ of the $12$ recommended physics books are selected?

Question

Here's the question.

A student is selecting $4$ out of $20$ recommended books for a certain course. Twelve are physics books. How many of these selections have exactly $3$ of the $12$ physics books?

Here's my set up.

$$\binom{20}{4}\binom{12}{8}$$

Is this the right way to do it?

Answer 1

Work to your constraints first. You are required to have only combinations that have three physics books in them: start with that; how many ways are there to choose three physics books?

Now, for each combination of 3 physics books, you have to choose a non-physics book. (Why? Because the question says "exactly" three physics books. If it said "at least three," then you need a different strategy.) How many non-physics books are there?

Pair the "how many ways to choose a single non-physics book" with each unique combination of "how many ways to pick three physics books" and you get your answer.