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Permutation and Combination Calculator

Last updated: April 2026

Permutations

Permutations count ordered arrangements. Order matters, so ABC and ACB are different outcomes.

P(n,r)

60

P(5,3) ordered arrangements

n!

120

(n-r)!

2

Downward product

5 x 4 x 3

Step-by-step working
  1. P(n,r) = n! / (n-r)!.
  2. P(5,3) = 5! / (5-3)!.
  3. = 5! / 2!.
  4. = 120 / 2 = 60.
  5. Alternative method: 5 x 4 x 3 = 60.

Permutations vs combinations

Use permutations when order matters, such as arranging letters, race finishing positions, PINs, or schedules. Use combinations when order does not matter, such as choosing a committee or lottery numbers.

Real-world examples

Permutations answer questions like how many ways three medals can be awarded from ten runners. Combinations answer questions like how many teams of three can be chosen from ten people.

Pascal's triangle and combinations

Each row of Pascal's triangle contains combination values. Row n gives C(n,0), C(n,1), C(n,2) and so on, with each value formed by adding the two values directly above it.

What does this mean?

This calculator is designed to help you understand the likely number before you make a decision or start an application.

Your result should be checked against official UK guidance, especially if your circumstances include dependants, exemptions, prior leave, or a complex immigration history.

Treat the figure as a planning tool rather than legal advice. Where the answer affects an application deadline or major payment, speak to an authorised adviser.

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