Permutation Formula

The Formula

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

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

Quick Example

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

Notation

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

What This Formula Means

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.

Formal View

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

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?

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

Why This Formula Matters

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

Frequently Asked Questions

What is the Permutation formula?

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)!}.

How do you use the Permutation formula?

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

What do the symbols mean in the Permutation formula?

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

Why is the Permutation formula important in Math?

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

What do students get wrong about Permutation?

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

What should I learn before the Permutation formula?

Before studying the Permutation formula, you should understand: factorial.

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Related Concepts

FactorialCombination