ProbabilityProbabilityHigh school

Counting

Permutations and combinations.

Counting principles answer how many ways something can happen without listing every case. The multiplication principle says that if one choice has $m$ options and a second has $n$ , there are $mn$ combined outcomes. Factorials, $n! = n(n - 1)\cdots 1$ , count the ways to arrange $n$ distinct objects in a row.

The key question is whether order matters. Permutations count ordered selections, $P(n, r) = \frac{n!}{(n - r)!}$ ; combinations count unordered selections, $C(n, r) = \frac{n!}{r!(n - r)!}$ . Choosing a president and treasurer is a permutation; choosing a two-person committee is a combination.

Key terms

Combination: Unordered selections of r items from n.
$\binom{n}{r} = \frac{n!}{r!(n-r)!}$

Key ideas

Multiplication principle for sequences of independent choices.
Order matters: permutation; order does not matter: combination.
$C(n, r) = C(n, n - r)$ .
Arrangements with repeated letters: divide $n!$ by the factorial of each repeat count.

Worked example

How many 3-person committees can be chosen from 10 people, and how many ways to pick a president, secretary and treasurer?

Committee, order irrelevant: $C(10, 3) = \frac{10 \cdot 9 \cdot 8}{3!} = 120$ .
Officers, order matters: $P(10, 3) = 10 \cdot 9 \cdot 8 = 720$ .
The ratio $720 / 120 = 6 = 3!$ counts orderings of each committee.

Answer: $120$ committees, $720$ officer slates

Common mistakes

Using permutations when order does not matter.
Adding instead of multiplying across independent stages.
Forgetting to divide by repeats when letters repeat.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.How many ways can 5 books be arranged on a shelf?
Show answer
$120$
2.Compute $C(8, 2)$ .
Show answer
$28$
3.How many distinct arrangements of the letters in LEVEL?
Show answer
$30$

Keep practising

Calculators

Permutation and Combination Calculator