# Tagged Questions

Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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### Proof by Induction (Inequality)

I'm given a inequality as such: $n! > n^3$ Where n > 5, I've done this so far: BC: n = 6, 6! > 720 (Works) IH: let n = k, we have that: $k! > k^3$ IS: try n = k+1, (I'm told to only work ...
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### Rearranging for $n$ with a factorial

For my maths course I need to prove that $n!/2^n$ tends to infinity as $n$ tends to infinity. For this I have to rearrange $n!/2^n > ∂$ so that it says $n > ...$. How to do this? Thanks in ...
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### How do I prove the equality of two sums?

I'm trying to prove the following: $$\sum_{r=0}^{n+1} \frac{n! (n+1)}{r!(n-r)!(n-r+1)} = \sum_{r=0}^n \frac{2n!}{r!(n-r)!}$$ I'm pretty sure that these are equal. How could I go about proving it? ...
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### Help solving an induction problem…$2^n < n!$ [closed]

Prove the following by induction. k and n in N (natural number) $n^k < 2^n$ $2^n < n!$ Need hint and help please. Thank you very much.
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### Limits and factorials

There is the following limit, I would like to calculate: $$\lim_{n\rightarrow\infty} \frac{2n}{(n!)^{1/n}}$$ After the substituion with Stirling's approximation I have got a relatively complex ...
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### Finding value of unknown in factorial equation

What's the value of $n$ in the following equation? $$2(2n-4)!= (n-4)!(n+2)!$$ I've tried coming up with an equivalent combination and expanding $(n+2)!$ but that didn't get me anywhere.
### What is the significance of $0!=1$?
One can derive $0!=1$ by the formula $\frac{n!}{(n-1)!}=n$ simply put $n=1$ and we get $0!=1$ . But a question remains in my mind: what is its physical significance? For example: $2!$ means: in ...
Let $n$ be a positive integer. In how many ways can one write $n!$ as a product of consecutive integers? For example: $4!=1\times2\times3\times4=2\times3\times4$. Here, $2$ possibilities exist. $... 1answer 75 views ### Does the equality$ \dfrac{10!}{6!} = 7! \$ hold any special (geometric) meaning?
I've come across the following simple, but unexpected equality numerous times accidentally. $$\frac{10!}{6!} = 7!$$ which is the same as $$1*2*3*4*5*6*7 = 7*8*9*10$$ Does it hold any specific (...