is 5 factorial equal to 120 related to any interesting mathematical patterns or just a coincidence

Question

Since the meme showed 5!=120, I started thinking about factorials.

0!=1
1!=1
2!=2
3!=6
4!=24
5!=120
6!=720
7!=5040
8!=40320
9!=362880
10!=3628800

I noticed:
- 5!=120 and 6!=720. The ratio 720/120=6, which makes sense since n!=n×(n−1)!.
- 10! seconds is exactly 6 weeks! (Since 10!=3,628,800 seconds =42 days)
- There are exactly 9! minutes in 6 weeks too? No wait...

Are there any other interesting patterns or coin

Alex Kim · Answer

For this kind of problem, I would write down exactly what is being counted before choosing a formula.

If order matters, you are usually in permutations or arrangements. If order does not matter, combinations are more likely. For induction, make the induction statement strong enough to actually imply the next step.

The standard induction pattern is:

1. prove the base case;
2. assume the statement for n=k;
3. use that assumption to prove it for n=k+1.

The third step must use the assumption

Is 5 factorial equal to 120 related to any interesting mathematical patterns or just a coincidence

1 Answer