MathematicsChapter 6: Permutation and Combination

PERMUTATIONS nPr

A permutation is an arrangement (order matters).

Learning objectives

State the main ideas and notation used in permutations npr.
Apply the rules of permutations npr in a clear, step-by-step solution.
Check signs, conditions, and the final answer for mathematical accuracy.

Definition

A permutation is an arrangement (order matters). The number of ways to arrange r objects taken from n different objects is nPr = n! / (n-r)! = n(n-1)...(n-r+1). nPn = n!. Meaning: first slot n choices, second n-1, ... r slots. If order does not matter, that is a combination (later).

Mathematical form

nPr = n

nPn = n

Where / symbols

PERMUTATIONS nPr: A permutation is an arrangement (order matters). The number of ways to arrange r objects taken from n different objects is nPr = n! / (n-r)! = n(n-1)...(n-r+1). nPn = n!. Meaning: first slot n choices, second n-1, ... r slots. If order does not matter, that is a combination (later).
1. FORMULA: 5P2 = 5*4 = 20. 5P5=120. 5P0=1 (empty arrangement). Need r <= n. Proof: n choices, then n-1, ... r factors.
2. WORDING: 'Arrange', 'line up', 'first, second, third', 'codes', 'passwords order' -> permutation. President, VP, secretary from 10 people: 10P3.

Relationship

nPr = n! / (n-r)!

Overview

Formula, then a lineup:

Formula

5P2 = 5*4 = 20. 5P5=120. 5P0=1 (empty arrangement). Need r <= n. Proof: n choices, then n-1, ... r factors.

Examples nP1 = n · nPn = n! · nP0 = 1

Wording

'Arrange', 'line up', 'first, second, third', 'codes', 'passwords order' -> permutation. President, VP, secretary from 10 people: 10P3.

Examples Gold silver bronze · Different positions · Not a committee yet

WHAT — PERMUTATIONS nPr

RULE — 1. FORMULA

5P2 = 5*4 = 20. 5P5=120. 5P0=1 (empty arrangement). Need r <= n. Proof: n choices, then n-1, ... r factors.

METHOD — 2. WORDING

'Arrange', 'line up', 'first, second, third', 'codes', 'passwords order' -> permutation. President, VP, secretary from 10 people: 10P3.

WORKED EXAMPLE

Worked: 7P3. 7P3 = 7*6*5 = 210 or 7!/(4!) = 504024 = 210

How to answer a permutations npr question

Identify the given quantities, notation, and required result.
Choose the matching rule and write it before substitution.
Work one algebraic step at a time, keeping brackets and signs clear.
Check restrictions, simplify the result, and verify it where possible.

Worked example

7P3.
7P3 = 7*6*5
= 210
or 7!/(4!) = 504024 = 210

In simple words

nPr needs r <= n. 5P7 is not a school permutation of distinct objects.

Key points

order matters
nPr = n! / (n-r)!
r factors falling
order words

Exam preparation

Exam Focus

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