# Which is larger? 20!20! or 2402^{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.

It is probably easier to note that $2^{40} = 4^{20}$. The only ones of the 20 factors in $20!$ that are smaller than $4$ are $1$, $2$ and $3$. But, on the other hand, $18$, $19$ and $20$ are all larger than $4^2$, so we can see

by comparing factor by factor.