# Sum of factorial as a sum of other factorials

I was doing some problem on factorial. Then I was striked by an hypothesis....

The hypothesis was....

Is it possible to show a factorial as a sum of other factorials???

Like

a! = b! + c! + d!.......

All numbers are unique Natural number without repetition.

I have tried to find out some solution but I was unable to find any!

Can anyone will try this as a challenge and give me the solution !!!

• Does this count? $$2!=1!+0!$$ – projectilemotion Apr 15 '17 at 17:08
• Sorry forgot to tell all should be natural number – Creepy Creature Apr 15 '17 at 17:41
• Some definitions have $0\in \mathbb{N}$ such as the standard ISO 80000-2. – projectilemotion Apr 15 '17 at 17:43

## 1 Answer

Apart from the trivial example $2!=1!+0!$, the answer is "no".

This follows from $$0!+1!+2!+3!+\ldots+(n-1)!\le n\dot (n-1)!=n!$$ with equalitiy iff all summands are equal to $(n-1)!$, i.e., $(n-1)!=1$.

• I'm not sure if I understand how it follows from that. – mrnovice Apr 15 '17 at 17:50
• How?? Any examples – Creepy Creature Apr 15 '17 at 18:08
• I believe he's saying that even if you sum all factorials less than a given factorial the sum is too small. In other words the factorial grows too fast. – Χpẘ Apr 15 '17 at 19:00