Prove $n!\mid\prod_{k=i}^{i+n-1}k$

Question

I have no idea how to prove this, I haven't yet learned much about this kind of product.

$$ n!\mid\prod_{k=i}^{i+n-1}k $$

Answer 1

Hint a very interesting fact you can use :

$$\dbinom {i+n-1} {i-1}=\frac{\cdots }{n!}$$

Answer 2

Notice that this relates strongly to the so-called choose function, $C(n,k)$, which is defined as $$C(N,k) = \frac{N!}{k!(N-k)!}$$ Since the right-hand side can be written $ \prod_{k=i}^{i+n-1} k = \frac{(i+n-1)!}{(i-1)!} $, the question is really asking you to verify that $$C(n+i-1, n) = \frac{(i+n-1)!}{n!(i-1)!} \in \mathbb Z$$