Permutation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Permutation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A permutation is an ordered arrangement of objects — the number of ways to choose and order r items from n distinct items is P(n,r) = \frac{n!}{(n-r)!}.

With permutations, order matters — first place and second place are different. Think of ranking students: ABC and BAC are different orderings.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Permutations count ordered selections: choose who goes first, then second, then third, multiplying the shrinking number of choices at each step.

Common stuck point: Permutation: 'How many ways to arrange?' Combination: 'How many ways to choose?'

Sense of Study hint: Ask: does switching the order create a different result? If yes, use permutations. Count choices for each slot and multiply.

Worked Examples

Example 1

easy
In how many ways can 3 students be arranged in a line from a group of 7?

Solution

  1. 1
    Recall the permutation formula for ordered selections: P(n, r) = \frac{n!}{(n-r)!}, where n = 7 and r = 3.
  2. 2
    Expand the factorial ratio: P(7, 3) = \frac{7!}{(7-3)!} = \frac{7!}{4!} = 7 \times 6 \times 5
  3. 3
    Calculate the product: 7 \times 6 \times 5 = 210

Answer

P(7,3) = 210
Permutations count ordered arrangements. Since the order in a line matters (ABC differs from BAC), we use permutations rather than combinations.

Example 2

medium
How many distinct arrangements of the letters in the word MISSISSIPPI are there?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
How many 4-digit PIN codes can be formed using the digits \{1,2,3,4,5,6\} if no digit may be repeated?

Example 2

medium
A club of 8 students must choose a president, vice president, and secretary. In how many ways can the officers be chosen?
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Related Concepts

FactorialCombination

Background Knowledge

These ideas may be useful before you work through the harder examples.

factorial