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.

Solution 1:

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 $$ 20! = 1\cdot 2\cdot3\cdots 18\cdot19\cdot 20 > \underbrace{1\cdot1\cdot1}_{3\text{ ones}}\cdot\underbrace{4\cdot4\cdots 4\cdot 4}_{14\text{ fours}}\cdot\underbrace{16\cdot16\cdot 16}_{3\text{ sixteens}} = 2^{40} $$ by comparing factor by factor.

Solution 2:

You can simply compute both numbers and compare them: $$ 20! = 2432902008176640000 \\ 2^{40} = 1099511627776 $$

So you can easily see that $2^{40} < 20!$