How many positive integers less than 100 are the product of three distinct primes? [ ]?

Here are your usable prime numbers:
1 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 

1x2x47 = 94, hence there are 14 combos with 1x2xn.
1x3x31 = 93, therefore there are only 9 distinct prime combinations here.
1x5x19 = 95, ergo there are only 5 qualifying combos here.
1x7x13 = 91, so here's 2 more possible combinations.

So there are only 30 combinations using the prime #1 in this particular scenario.  Now to start with the only even prime #, #2 to find more combos that suffice these criteria.

2x3x13 is the last one since 2x3x17 = 102.
Therefore there are only 4 possible combos here.
2x5x7 is the only combo that works for the 2x5xn model, so add 1 more.
That is the last 2xnxn model that will work.  Now for 3xnxn

3x5x7 yields 105, therefore, it is logically and theoretically impossible for another nxnxn combo to exist ≤ 100.

So to recap this:
1xnxn = 30 combos
2xnxn = 5 combos
3xnxn and higher had 0 combos (for increasing n values).

ANSWER: 35 different COMBOS !!!