Permutation and Combination Calculator

🔢 Permutations (order matters)

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🎯 Combinations (order doesn't matter)

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What Are Permutations and Combinations?

Permutations count the number of ways to arrange r items from a set of n, where order matters. Combinations count the number of ways to choose r items from n, where order doesn't matter. Both are core concepts in probability and combinatorics.

How to Use This Calculator

  1. Enter the total number of items (n).
  2. Enter how many items you're selecting or arranging (r).
  3. View both the permutation and combination counts instantly.

Permutation and Combination Formulas

Permutations
P(n,r) = n! / (n−r)!
Combinations
C(n,r) = n! / [r! × (n−r)!]

Worked Example

Example: n = 10, r = 3

Permutations: 10! / (10−3)! = 10 × 9 × 8 = 720

Combinations: 720 / 3! = 720 / 6 = 120

Understanding Your Results

Use permutations when order matters — like arranging 3 people in a race for 1st, 2nd, and 3rd place. Use combinations when order doesn't matter — like choosing 3 people for a committee, where their selection order is irrelevant.

When to Use Each

  • Permutations: passwords with distinct characters, race rankings, seating arrangements, any scenario where sequence matters.
  • Combinations: lottery number selections, committee formation, card hands, any scenario where only the group selected matters, not the order chosen.

Common Mistakes to Avoid

  • Using the permutation formula when order genuinely doesn't matter (over-counting duplicate arrangements).
  • Using the combination formula when order does matter (under-counting valid arrangements).
  • Attempting factorials with very large n, which quickly produce astronomically large numbers.

Frequently Asked Questions

Permutations count arrangements where order matters (like race rankings); combinations count selections where order doesn't matter (like committee membership).

Use P(n,r) = n! / (n−r)!, where n is the total number of items and r is the number being arranged.

Use C(n,r) = n! / [r! × (n−r)!], which is the permutation formula divided by r! to remove duplicate orderings.

Because combinations don't care about order, each unique group of r items has r! different possible orderings that would all count as the same combination — dividing by r! removes this over-counting.

n! means multiplying all positive integers from 1 up to n. For example, 5! = 5×4×3×2×1 = 120. By convention, 0! = 1.