Permutation Calculator — Definition, Formula & Examples

A permutation calculator computes how many ways you can arrange a chosen number of items from a larger set when the order matters. You enter the total number of items

n

and the number you are choosing

r

, and it returns

P(n,r)

The permutation function $P(n,r)$ , also written $_nP_r$ , equals $\dfrac{n!}{(n-r)!}$ and counts the number of injective functions from an $r$ -element set into an $n$ -element set, or equivalently, the number of distinct ordered $r$ -arrangements of $n$ distinct objects.

Key Formula

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

Where:

$n$ = Total number of distinct items in the set
$r$ = Number of items being arranged (chosen and ordered)
$!$ = Factorial — the product of all positive integers up to that number

How It Works

Enter the total number of items

n

and the number you want to arrange

r

. The calculator evaluates

n!

divided by

(n-r)!

. Because the factorials cancel, you only need to multiply the

r

largest factors of

n!

: that is,

n \times (n-1) \times \cdots \times (n-r+1)

. This shortcut keeps the arithmetic manageable even for large values of

n

Worked Example

Problem: A club of 10 members elects a president, vice-president, and treasurer. How many ways can these 3 positions be filled?

Identify n and r: There are 10 members and 3 ordered positions, so

n = 10

and

r = 3

Apply the formula: Substitute into the permutation formula.

P(10,3) = \frac{10!}{(10-3)!} = \frac{10!}{7!}

Simplify: Cancel

7!

from numerator and denominator, leaving only the top three factors of

10!

10 \times 9 \times 8 = 720

Answer: There are 720 ways to fill the three positions.

Why It Matters

Permutation calculations appear constantly in probability courses and on standardized tests like the SAT and ACT. They also underpin real-world problems in scheduling, cryptography, and any scenario where the sequence of choices matters.

Common Mistakes

Mistake: Confusing permutations with combinations and dividing by

r!

when order matters.

Correction: Only divide by

r!

when the order does NOT matter (combinations). If each different ordering counts as a distinct outcome — like assigning ranked positions — use

P(n,r)

without the extra division.

Related Terms

Combinatorics — The broader field covering permutations and combinations
Factorial — The core operation inside the permutation formula