What is the value of n for which (3!)(5!)(7!)=n!?

Question

Please explain your answer thoroughly!

Answer 1

3! = 3 x 2 x 1
5! = 5 x 4 x 3 x 2 x 1
7! = 7 x 6 x 5 x 4 x 3 x 2 x 1

Now just start creating the digits starting with 1:
1, 2, 3 come from 3!
4, 5 come from 5!
6, 7 come from 7!

That leaves 3 x 2 x 1 (from 5!) and 5 x 4 x 3 x 2 x 1 (from 7!)

We can make 8 from 4 x 2
We can make 9 from 3 x 3
We can make 10 from 2 x 5

That leaves nothing but ones, so we are done.

10! = (2 x 5)(3 x 3)(4 x 2) x 7 x 6 x 5 x 4 x 3 x 2 x 1

Answer 2

3!5!7! = 1*2*3*4*5*6*7*(2*4)*(3*3)*(2*5) = 10!
n = 10