permutations calculator

What is a permutation?

A permutation is an arrangement where order matters. If you pick first, second, and third place winners, that order is important. In contrast, combinations ignore order.

For example, choosing letters A, B, and C:

• Combination: {A, B, C} is one group.
• Permutation: ABC, ACB, BAC, BCA, CAB, and CBA are all different outcomes.

Permutation formulas

1) Without repetition

Use this when items cannot repeat and each item is used at most once:
nPr = n! / (n - r)!

Example: from 10 people, how many ordered groups of 3 can you form?
10P3 = 10 × 9 × 8 = 720

2) With repetition

Use this when each position can be filled by any of the n items, including repeats:
n^r

Example: 4-digit PIN using digits 0–9:
10^4 = 10,000

How to use this permutations calculator

• Select the permutation type (with or without repetition).
• Enter n as the total available items.
• Enter r as the number of positions or chosen items.
• Click Calculate Permutations.

The calculator returns the exact result (using big-integer arithmetic) plus a quick formula summary.

When should you use permutations?

• Assigning ranked positions (1st, 2nd, 3rd).
• Arranging books, people, or tasks in order.
• Creating ordered passcodes or sequences.
• Modeling outcomes in probability and statistics.

Common mistakes to avoid

Confusing permutations and combinations

If order matters, use permutations. If order does not matter, use combinations.

Using invalid values

For no-repetition permutations, r cannot be larger than n.

Forgetting the repetition rule

If repeats are allowed, use n^r; otherwise use nPr.

Quick examples

Example A: Without repetition

You have 8 runners and want to count gold, silver, and bronze outcomes:
8P3 = 8 × 7 × 6 = 336.

Example B: With repetition

You form a 5-character code from 26 uppercase letters with repeats allowed:
26^5 = 11,881,376.

Final takeaway

A permutations calculator saves time and prevents arithmetic errors, especially for large values. The key is choosing the correct model: use nPr when order matters without repeats, and n^r when repeats are allowed.