Order matters, repetition not allowed

How many ways can we select $k$ persons from a group of $n$ people to stand in line for a photo shoot? There are:

• $n$ ways to select the first person
• $n-1$ ways to select the second person
• and so on until the $n-(k-1)$ ways to select the $k$th person

So, using the product rule, we find that:

$$\begin{array}{rl}\boxed{n\cdot (n-1)\cdot ...\cdot (n-k+1)}& =\frac{n\cdot (n-1)\cdot ...\cdot (n-k)\cdot ...\cdot 2\cdot 1}{1\cdot 2\cdot ...\cdot (n-k)}\\ & =\frac{n!}{(n-k)!}\end{array}$$

There is a special case when $k=n$:

$$\begin{array}{rl}\frac{k!}{(k-k)!}& =\frac{k!}{0!}\\ & =\frac{k!}{0}\\ & =\boxed{k!}\end{array}$$