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.

You can also separate each one of these, into 10 terms and note:
$$1\times 20 > 2^4$$
$$2\times 19 > 2^4$$
$$\vdots$$
$$10 \times 11 >2^4$$
$$\Rightarrow 20! > 2^{40}$$
The idea is to break the factorial symmetrically into smaller pieces; which is not the most robust method for inequalities which include factorials; but it is useful in certain obvious cases.
