How is this binomial coefficient be written in this way?

Question

$$\begin{pmatrix} p\\ n\end{pmatrix}=\frac{p!}{n!(p-n)!} = \frac{p(p-1)(p - 2) \dots (p-(n-1))}{n!}$$

I was reviewing and going over the binomial theorem. While doing that, I came across this binomial coefficient and began to feel confused about how it simplifies to the result on the right-hand side. What is the process behind this simplification?

Answer 1

Simplify $p !$ with $(p - n)!:$ $$\frac{p!}{(p-n)!} = \frac{(p - n)! (p - n + 1) \dots (p - 1) p}{(p - n)!} = (p - n + 1) \dots (p - 1) p,$$ which yields the result.

Answer 2

See, leaving out the n! in the denominator for now

$$\begin{split}\frac{p!}{(p-n)!}&=\frac{p*(p-1)*(p-2)*...*(p-n+1)*(p-n)*(p-n-1)*...*2*1}{(p-n)*(p-n-1)*...*2*1}\\ &=p*(p-1)*(p-2)*...*(p-(n-2))*(p-(n-1))\end{split}$$

Divide by the original n! to achieve

$$\frac{p*(p-1)*(p-2)*...*(p-(n-2))*(p-(n-1))}{n!}$$

Basically, it truncates p! to the first p - (p-(n-1)) + 1 terms