Permutations

A permutation is an ordered arrangement of objects. Order matters!

All n Objects

The number of ways to arrange $n$ distinct objects:

$$P(n) = n! = n \times (n-1) \times \cdots \times 2 \times 1$$

Example: Arranging 3 books on a shelf: $3! = 6$ ways.

k Objects from n

Choosing $k$ objects from $n$ and arranging them in order:

$$P(n, k) = \frac{n!}{(n-k)!}$$

Example: Selecting president, VP, secretary from 10 people:

$$P(10, 3) = \frac{10!}{7!} = 10 \times 9 \times 8 = 720$$

Why Division?

We divide by $(n-k)!$ because we’re only arranging $k$ objects, not all $n$.

Think of it as:

• Choose position 1: $n$ choices
• Choose position 2: $n-1$ choices
• Choose position k: $n-k+1$ choices
• Total: $n \times (n-1) \times \cdots \times (n-k+1) = \frac{n!}{(n-k)!}$

Example: Race Rankings

In a race with 8 runners, how many ways can gold, silver, bronze be awarded?

$$P(8, 3) = \frac{8!}{5!} = 8 \times 7 \times 6 = 336$$