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Learn · Mathematics · Grade 10 · Counting

Counting: permutations and combinations

The multiplication rule, factorials, arrangements where order matters (P(n, r)) and selections where it does not (C(n, r)), with the "committee or queue?" test for telling them apart.

High school Lesson 14 of 31 Interactive figure Practice

First, the rules

Multiply, then decide whether order matters

★ Multiplication principle
Example 1
If a task has m ways and a second, independent task has n ways, together they have m · n ways.
Addition principle
"This OR that" (no overlap): add the counts.
Factorial
Example 2
n! = n · (n − 1) · … · 2 · 1, 0! = 1
n! arranges n different objects in a row.
Repeated objects in a row
Example 2
n!p! · q! · …
Divide by the factorial of each repeated group (the letters of BANANA: 6! / (3! · 2!)).
★ Permutation — order matters
Example 3
P(n, r) = n!(n − r)! = n(n − 1)…(n − r + 1)
A queue, a podium, a code, a word.
★ Combination — order does not matter
Example 3
C(n, r) = n!r! (n − r)!
A committee, a handshake, a hand of cards, a subset. C(n, r) = C(n, n − r).
★ The test
Example 3
Swap two of the chosen items. Did you get something different? Then order matters: permutation. Same thing? Combination.
Around a circle
(n − 1)! — rotations are the same seating, so one person is "fixed".

See it

Grow the tree

Each choice adds a level to the tree; the leaves are the outcomes. Below it, fill r slots from n objects and watch the falling product n · (n − 1) · … become P(n, r), then C(n, r) after dividing by r!.

Step by step

Worked examples

1

Outfits and codes

multiplication principle

(a) A student has 4 shirts, 3 pairs of trousers and 2 pairs of shoes. How many outfits? (b) How many 4-digit codes can be made from the digits 1–9 with no digit repeated? With repetition allowed?

Ask yourself: One decision per slot: how many choices does each slot have?

Show the solution
  1. (a) 4 · 3 · 2 = 24 outfits.
  2. (b) No repeats: the slots have 9, 8, 7, 6 choices: 9 · 8 · 7 · 6 = 3024 (this is P(9, 4)).
  3. With repetition every slot has 9 choices: 94 = 6561.

Answer 24; 3024 and 6561

2

Arranging letters

factorial · repeated letters

How many different "words" (any order, no meaning needed) can be made from the letters of (a) MATH, (b) LEVEL?

Ask yourself: Are any letters repeated? Each repeated group divides the count.

Show the solution
  1. (a) Four different letters: 4! = 24.
  2. (b) LEVEL has 5 letters with L twice and E twice: 5!2! · 2! = 1204 = 30.

Answer 24 and 30

3

Podium versus committee

permutation vs combination

Eight runners. (a) In how many ways can gold, silver and bronze be awarded? (b) In how many ways can 3 of them be chosen for a drug test?

Ask yourself: Swap two of the chosen runners. In (a) the medals change hands — different result. In (b) the same three are tested.

Show the solution
  1. (a) Order matters: P(8, 3) = 8 · 7 · 6 = 336.
  2. (b) Order does not matter: C(8, 3) = 8 · 7 · 63! = 3366 = 56.
  3. Notice (b) = (a) / 3!: each committee of 3 corresponds to 3! = 6 orderings.

Answer 336 and 56

Practice: counting

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Outfits, podiums, committees, round tables and words with repeated letters — type the count. Your best score is kept in this browser only.

About this lesson

Counting for high school: the addition and multiplication principles, factorial, permutations P(n, r) for arrangements in order, combinations C(n, r) for selections, arrangements with repeated letters, circular arrangements, and the one question that decides which formula to use. Pascal’s triangle and the symmetry C(n, r) = C(n, n − r). Worked examples and practice questions with explanations.

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