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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?

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    @ArturoMagidin - in your very first comment you sought a _valid_ clarification. You observed that "`numbers that are the product of exactly 3 distinct primes less than a 100` ... [is] `a bit trickier` [problem]". My question is about the **bit** bit. :) See http://math.stackexchange.com/questions/1733791/. I do not think it is trivial, esp as the number becomes much larger than 100.2017-04-06

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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?

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    @Gregslu: What exactly is tricky about it? You have 25 things to choose from, you want to choose three of them, the answer is $\binom{25}{3}$. No tricks at all.2011-12-13