Which is larger? $20!$ or $2^{40}$? Someone asked me this question, and it bothers the hell out of me that I can't prove either way.
I've sort of come to the conclusion that 20! must be larger, because it has 36 prime factors, some of which are significantly larger than 2, whereas $2^{40}$ has only factors of 2.
Is there a way for me to formulate a proper, definitive answer from this?
Thanks in advance for any tips. I'm not really a huge proof-monster.

Here's my take on this:
First find the number of factors of 2 in 20! This can be done with
$\left\lfloor \frac{20}{2^{1}} \right\rfloor\; \; +\; \left\lfloor \frac{20}{2^{2}} \right\rfloor\; +\; \left\lfloor \frac{20}{2^{3}} \right\rfloor\; +\; ...\; +\; \left\lfloor \frac{20}{2^{5}} \right\rfloor\; \; \; $
Computing this you get there are 18 factors of 2.
Next, find the factors of 3. Using the same method as above except taking powers of 3 instead of powers of 2 you get that there are 8 factors of 3. Each factors of 3 is about 1.5 factor of 2 because $\log_2(3) = 1.5$ so it is like there are 8*1.5 = 12 additional factors of 2.
Next do the same for factors of 5. There are 4 factors of 5, but each factor of 5 is about 2.3 factors of 2 (from $\log_2(5)$) so it is like there are 4*2.5 = 10 extra factors of 2.
In total there are now 18 + 12 + 10 = 40 factors of 2. There will be even more as you can also take 7, 11, 13, 17, and 19 as primes, so it is now clear that 20! is much larger.
