It is said that factorial grows faster than exponential, but in the case of permutation:
permutation with repetition = n^n
permutation without repetition = n!
And of course permutation with repetition has a bigger "space" than permutation without repetition which means it should grow faster.
I was looking for some convincing proof that shows this in an intuitive way.
Like for example displaying n! in a n^0.n^(m-1).n^(m-2)…n^1 and eventually shows that the sum of the exponential is < n, but did not succeed.
I am not sure if anyone can help me with the proof or maybe just help to explain the case against the "factorial grows faster than exponential" fact
Sorry for my wording, I have very poor mathematical backgrounds
Best Answer
The $n^n$ function is not an exponential as the basis varies.
To prove $n!=o(n^n)$, let's evaluate the ratio: $$\frac{n!}{n^n}=\frac1n\cdot\frac2n\cdots\frac{n-1}n\cdot \frac nn <\frac 1n$$ since each factor $\dfrac kn$ is less than $1$ if $k<n$. This inequality implies $\;\displaystyle\lim_{n\to\infty}\frac{n!}{n^n}=0$.