Undoing Capital Pi

Question

I have an equation of which I'm trying to undo capital pi notation to simply get the expression. The equation is

$c=\prod_{i=1}^k n-(i-1)$

How would I undo this?

Answer 1

What do you mean by 'undo'?   Do you mean 'unpack', or 'expand'?

The $\prod$ notation is a product series (that is, it is the product of terms of a sequence).

So expanding this is simply:

$$\begin{align}c & =\prod^k_{i=1} \big(n−(i−1)\big) \\[1ex] & = (n-1+1)\cdot(n-2+1)\cdot(n-3+1)\cdots(n-k+1) \\[1ex] & =\frac{n!}{(n-k)!}\end{align}$$

That is all.