Unique factorization less than 100

Question

How do I approach this problem using unique factorization?...

How many numbers are product of (exactly) $3$ distinct primes $< 100$?

edit: Just to add to that, How does unique factorization into primes play an important role in answering this question?

Answer 1

So, you want to count the number of integers $n$ that can be written as $$n = pqr,$$ where $p$, $q$, and $r$ are pairwise distinct primes, each less than a 100. By unique factorization, it suffices to first pick the three distinct primes and then multiply them together.

You've counted that there are 25 primes less than 100. You want to pick three of them. How many ways are there to do so?