How to divide a factorial by an exponent...

Great question - factorial problems that involve division almost always have the same key to solving them - you have to factor.

Consider this question:

What is the greatest value y for which 3^y is a factor of 10!?

The key is to recognize that the question is really asking "how many times can 10! be evenly divided by 10!", or in other words "how many 3s are embedded within 10!"

Here you want to break out 10! to find where the 3s are. 10! is:

10*9*8*7*6*5*4*3*2*1

Many of these individual components won't have any 3s in them (10, for example, is not divisible by 3), but the 9, the 6, and the 3 will each give you factors of 3. But be careful - to isolate the 3s you'll need to factor out those numbers:

3 = 3
6 = 3*2
9 = 3*3 (you actually get two factors of 3 from that 9)

So the answer is 4 - you can divide 10! by 3 four times.

And more important is the takeaway - when you're asked to divide a factorial by an exponent, break the exponent's base into its prime factors, and then break out the factorial so that you can see each occurrence of that prime factor. And remember - one of the GMAT's favorite tricks is to give you a factorial that includes the square of the number you're factoring (in this case you were looking for 3s, and 9 was in the factorial).


Hope that helps...