Why does $\binom{2n}{n} = \prod_{i=1}^n (\frac{n+i}{i} )$

Question

I cannot understand why are you able to generalize this like this

$$\binom{2n}{n} = \frac{2n × (2n−1) × ... × (n+ 2) × (n+ 1)}{n ×(n−1)×...×2×1} =\prod_{i=1}^n (\frac{n+i}{i}) $$

I get that $$\binom{2n}{n} = \frac{2n!}{n! (n-n)!} $$ and therefore this is correct $$\frac{2n × (2n−1) × ... × (n+ 2) × (n+ 1)}{n ×(n−1)×...×2×1} $$

However, how do you get to that product generalization?

What would you suggest I read to understand it?

Answer 1

We simply have

$$\frac{2n × (2n−1) × ... × (n+ 2) × (n+ 1)}{n ×(n−1)×...×2×1}=\frac{2n}{n} × \frac{2n−1}{n-1} × \frac{2n−2}{n-2}× \ldots × \frac{n+1}{1}= \prod_{i=1}^n \left(\frac{n+i}{i}\right)$$