Permutation & Combination Calculator

Calculate permutations P(n,r) and combinations C(n,r) with step-by-step solutions and visualizations.

Calculator

Number of total items (must be ≥ 0)

Number of items to select/arrange (must be ≥ 0 and ≤ n)

Select calculation type (Permutation or Combination) and enter n and r values to calculate with step-by-step solutions.

Complete User Guide

What are Permutations and Combinations?

Permutations and combinations are ways to count arrangements and selections of items from a set.

  • Permutation P(n,r): Number of ways to arrange r items from n items where order matters
  • Combination C(n,r): Number of ways to select r items from n items where order does not matter

Formulas

Permutation Formula

P(n, r) = n! / (n - r)!

Example: P(5, 3) = 5! / (5-3)! = 120 / 2 = 60

Combination Formula

C(n, r) = n! / (r! × (n - r)!)

Example: C(5, 3) = 5! / (3! × 2!) = 120 / 12 = 10

Key Differences

  • Permutations: Order matters (ABC is different from CBA)
  • Combinations: Order does not matter (ABC is the same as CBA)
  • Permutations are always greater than or equal to combinations for the same n and r

Examples

Example 1: Permutation

Problem: How many ways can 3 people be arranged from 5 people?

Solution: P(5, 3) = 60

Answer: 60 different arrangements

Example 2: Combination

Problem: How many ways can 3 people be selected from 5 people?

Solution: C(5, 3) = 10

Answer: 10 different selections

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