Permutation & Combination Calculator

Calculate permutations P(n,r) and combinations C(n,r) with factorial breakdown.

By Konstantin Iakovlev · Updated April 2026 · Source: Khan Academy

Permutations P(n,r)

720

Combinations C(n,r)

120

Formulas

P(10,3)10! / (10-3)! = 720
C(10,3)10! / (3!(10-3)!) = 120
10!3,628,800

Use the Permutation & Combination Calculator above to calculate your results. Enter your values and see instant results — all calculations run in your browser.

Disclaimer: This calculator is for informational purposes only and does not constitute tax, financial, or legal advice. Results are estimates based on the information you provide and current rates. Always consult a qualified tax professional or financial advisor for advice specific to your situation.

How It Works

Counting how many ways you can arrange or select items from a set is what this tool does, and quickly. Statisticians reach for it, data scientists modeling 2026 election probabilities depend on it, and students working through advanced probability use it to map out the possible outcomes of a scenario in seconds.

Permutations P(n,r) come from n! / (n-r)!, where 'n' is the total count of items and 'r' is how many get arranged. Combinations C(n,r) use n! / (r! * (n-r)!), the extra r! in the denominator stripping out arrangements that differ only in order. Both rest on factorials, written with the '!' symbol.

Order is the dividing line: it counts in permutations and is ignored in combinations, and blurring the two is the easiest way to go wrong. Confirm that 'n' is at least as large as 'r', and look over your inputs once more before trusting the output—especially as the values climb and the factorials grow large.

Example: Selecting a 2026 Olympic Squad

  1. 1 Imagine you're the coach for the 2026 Winter Olympics alpine skiing team. You have 15 highly qualified athletes (n=15) and need to select a squad of 4 to compete in the downhill event (r=4).
  2. 2 To find the number of different groups of 4 athletes you can select (order doesn't matter), you'd calculate C(15, 4). If you wanted to arrange 4 athletes for specific starting positions (order matters), you'd calculate P(15, 4).
  3. 3 Using C(15, 4) = 15! / (4! * (15-4)!) = 15! / (4! * 11!) = 1365. Using P(15, 4) = 15! / (15-4)! = 15! / 11! = 32760.
  4. 4 This means there are 1365 different combinations of 4 athletes you could choose for your squad, and 32760 different ways to arrange 4 athletes in specific starting positions. This demonstrates the significant difference when order is considered.

Source: Khan Academy · Last updated: April 2026

Frequently Asked Questions

What is the difference between a permutation and a combination?
Permutations care about order (ABC is different from BCA), while combinations do not (ABC and BCA are the same group). Use permutations for rankings, arrangements, and sequences. Use combinations for groups and selections.
How do you calculate permutations P(n,r)?
P(n,r) = n! / (n-r)!. For choosing and ordering 3 items from 10: P(10,3) = 10! / 7! = 10 x 9 x 8 = 720 possible arrangements.
How do you calculate combinations C(n,r)?
C(n,r) = n! / (r!(n-r)!). For choosing 3 items from 10 (order does not matter): C(10,3) = 10! / (3! x 7!) = 720 / 6 = 120 possible groups.

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