$\frac{N!}{(N-n)!}$ when $n<<N$

Question

I need to show that for $n<<N$ then $\frac{N!}{(N-n)!} \approx N^{n} $ I can see that $\frac{N!}{(N-n)!} = (N)(N-1)...(N-(n-1))$

and intuitively its clear but I am unable to show rigorously. What should be my approach ?

Answer 1

We have $$ \lim_{N \to \infty} \frac{N(N-1)\cdots(N-(n-1))}{N^n} = \lim_{N \to \infty} 1(1-1/N)\cdots(1-(n-1)/N)=1. $$