Permutations Calculator

What is a permutations calculator?

A permutations calculator tells you how many different ordered arrangements you can make by selecting $r$ items from a larger set of $n$ distinct items. Because order matters, picking item A then item B is counted separately from picking B then A.

Permutations show up whenever you need to count sequences: assigning gold, silver, and bronze medals to runners, choosing a president, vice‑president, and treasurer from a club, or working out how many distinct passwords or PIN orderings are possible.

How does it work?

Enter the total number of items $n$ and how many you want to arrange $r$. The calculator evaluates the standard permutation formula and returns the result instantly. It expects whole, non‑negative numbers, and requires $r \le n$ — you cannot arrange more items than you have.

The number of permutations of $r$ items taken from $n$ is:

${}^{n}P_{r} = \frac{n!}{(n-r)!}$

Here $n!$ (read “n factorial”) is the product of all positive integers up to $n$, and $0! = 1$ by definition. Unlike a combination, a permutation distinguishes between different orderings of the same selection.

Worked examples

• n = 5, r = 2. ${}^{5}P_{2} = \frac{5!}{3!} = \frac{120}{6} = 20$ ordered pairs.
• n = 10, r = 3. ${}^{10}P_{3} = \frac{10!}{7!} = 10 \times 9 \times 8 = 720$ arrangements.
• n = 5, r = 5. ${}^{5}P_{5} = \frac{5!}{0!} = 120$, which is simply $5!$ — every full ordering of all five items.
• n = 5, r = 0. ${}^{5}P_{0} = \frac{5!}{5!} = 1$, the single “empty” arrangement.

If you ask for $r > n$ — for instance $n = 3$ and $r = 5$ — the result is left blank, because there is no valid arrangement.

Practical notes

When order does not matter, you want a combination instead, which divides the permutation count by $r!$ to remove duplicate orderings. The building block of both is the factorial, and growth of these counts is closely related to repeated multiplication explored in the exponent calculator.

Because factorials grow very quickly, permutation counts can become enormous: ${}^{20}P_{20} = 20!$ already exceeds $2.4 \times 10^{18}$. For large $n$ the result is an approximation limited by floating‑point precision.