Permutation

Statistics
definition

Also known as: arrangement

Grade 9-12

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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)!}. Permutations count passwords, seating arrangements, race finishes, and any situation where the order of selection changes the outcome.

Definition

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)!}.

💡 Intuition

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

🎯 Core Idea

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

Example

Arrange 3 people in 3 chairs: 3! = 3 \times 2 \times 1 = 6 \text{ ways}

Formula

P(n, r) = \frac{n!}{(n - r)!}

Notation

P(n, r), _nP_r, or P^n_r all denote permutations of r items from n

🌟 Why It Matters

Permutations count passwords, seating arrangements, race finishes, and any situation where the order of selection changes the outcome.

💭 Hint When Stuck

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

Formal View

P(n, r) = \frac{n!}{(n-r)!} = n(n-1)(n-2)\cdots(n-r+1) for 0 \leq r \leq n

Related Concepts

FactorialCombination

🚧 Common Stuck Point

Permutation: 'How many ways to arrange?' Combination: 'How many ways to choose?'

⚠️ Common Mistakes

  • Using permutations when order does not matter — if selecting a committee, use combinations instead
  • Confusing P(n, r) = \frac{n!}{(n-r)!} with n^r — the latter allows repetition, permutations do not
  • Forgetting that P(n, n) = n! — arranging all n items uses factorial, not the permutation formula with r < n

Frequently Asked Questions

What is Permutation in Math?

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)!}.

Why is Permutation important?

Permutations count passwords, seating arrangements, race finishes, and any situation where the order of selection changes the outcome.

What do students usually get wrong about Permutation?

Permutation: 'How many ways to arrange?' Combination: 'How many ways to choose?'

What should I learn before Permutation?

Before studying Permutation, you should understand: factorial.

Prerequisites

factorial

Next Steps

combination

How Permutation Connects to Other Ideas

To understand permutation, you should first be comfortable with factorial. Once you have a solid grasp of permutation, you can move on to combination.

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Visualization

Static

Visual representation of Permutation