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 240 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 240=420. 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 42, so we can see
20!=1⋅2⋅3⋯18⋅19⋅20>1⋅1⋅1⏟3 ones⋅4⋅4⋯4⋅4⏟14 fours⋅16⋅16⋅16⏟3 sixteens=240
by comparing factor by factor.