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This can be treated as a combination question. The order of the prime factors does not matter (ex. 5 * 7 is the same as 7 * 5). And we know that no positive integers will be duplicated because the product of prime factors is unique.
We are looking for 4C4 + 4C3 + 4C2.
That is, given 4 items how many ways can we group 4 into a group of 4 (4C4) + how many ways to group 4 into a group of 3 (4C3) + how many ways to group 4 into a group of 2 (4C2).
This way we are finding all the possible combinations of 2 or more prime factors without repeating any.
The formula for nCr is n!/((n-r)!r!)
4C4 = 1, nCn always equals 1
4C3 = 4
4C2 = 6
Sum = 11
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