Skill: Calculate permutations without repetition

When you arrange $r$ distinct objects chosen from $n$ available objects without repetition, order matters. This is called a permutation.

Use the formula

$$nP_r=\frac{n!}{(n-r)!}$$

where $n!$ means factorial:

$$n!=n(n-1)(n-2)\cdots 1.$$

Example

A class has $8$ students. In how many ways can the teacher choose a president, a vice-president, and a secretary?

Because these are different positions, order matters. Choosing Ana as president and Ben as vice-president is different from choosing Ben as president and Ana as vice-president.

There are three positions to fill:

1. Choose the president: $8$ choices.
2. Choose the vice-president: $7$ remaining choices.
3. Choose the secretary: $6$ remaining choices.

By the multiplication principle,

$$8\cdot 7\cdot 6=336.$$

Using permutation notation gives the same result:

$$8P_3=\frac{8!}{(8-3)!} =\frac{8!}{5!} =\frac{8\cdot7\cdot6\cdot5!}{5!} =8\cdot7\cdot6 =336.$$

Therefore, there are

$$\boxed{336}$$

ways to assign the three positions.

Remember: use permutations when the objects are distinct, no object is used more than once, and changing the order creates a different outcome.