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) = n!/(n-r)!.

The Formula

P(n,r)=n!(n−r)!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!=3×2×1=6 ways3! = 3 \times 2 \times 1 = 6 \text{ ways}

Notation

P(n,r)P(n, r), nPr_nP_r, or PrnP^n_r all denote permutations of rr items from nn

What This Formula Means

A permutation is an ordered arrangement of objects — the number of ways to choose and order rr items from nn distinct items is P(n,r)=n!(n−r)!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)=n!(n−r)!=n(n−1)(n−2)⋯(n−r+1)P(n, r) = \frac{n!}{(n-r)!} = n(n-1)(n-2)\cdots(n-r+1) for 0≤r≤n0 \leq r \leq n

Worked Examples

Example 1

easy
In how many ways can 33 students be arranged in a line from a group of 77?

Answer

P(7,3)=210P(7,3) = 210

First step

1
Recall the permutation formula for ordered selections: P(n,r)=n!(n−r)!P(n, r) = \frac{n!}{(n-r)!}, where n=7n = 7 and r=3r = 3.

Full solution

  1. 2
    Expand the factorial ratio: P(7,3)=7!(7−3)!=7!4!=7×6×5P(7, 3) = \frac{7!}{(7-3)!} = \frac{7!}{4!} = 7 \times 6 \times 5
  2. 3
    Calculate the product: 7×6×5=2107 \times 6 \times 5 = 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?

Example 3

medium
Solve for nn: P(n,2)=72P(n,2) = 72.

Common Mistakes

  • Using a permutation when order doesn't matter — that overcounts by r!r!; use a combination instead.
  • Allowing repeats — P(n,r)P(n,r) assumes distinct items each used once; repeats need the counting principle.
  • Confusing nn and rr in n!(n−r)!\frac{n!}{(n-r)!} — nn is the pool, rr is how many positions you fill.

Why This Formula Matters

Permutations are half the counting decision that every probability problem depends on — get 'does order matter?' wrong and your sample-space size is off by a factor of r!r!. They distinguish a podium (gold-silver-bronze) from a committee. Recognizing it by "Does swapping two of the chosen items create a different valid outcome?" — rather than by familiar numbers — is what lets a student tell it apart from combination and factorial and counting principle in a mixed problem set.

Frequently Asked Questions

What is the Permutation formula?

A permutation is an ordered arrangement of objects — the number of ways to choose and order rr items from nn distinct items is P(n,r)=n!(n−r)!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)P(n, r), nPr_nP_r, or PrnP^n_r all denote permutations of rr items from nn

Why is the Permutation formula important in Math?

Permutations are half the counting decision that every probability problem depends on — get 'does order matter?' wrong and your sample-space size is off by a factor of r!r!. They distinguish a podium (gold-silver-bronze) from a committee. Recognizing it by "Does swapping two of the chosen items create a different valid outcome?" — rather than by familiar numbers — is what lets a student tell it apart from combination and factorial and counting principle in a mixed problem set.

What do students get wrong about Permutation?

The procedure for permutation is the easy part; the trap is using a permutation when order doesn't matter. Asking "Does swapping two of the chosen items create a different valid outcome?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Permutation formula?

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

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Related Concepts

FactorialCombination