About Permutation & Combination Calculator
Our Permutation and Combination Calculator is a comprehensive tool for solving P(n,r), C(n,r), and factorial calculations. Whether you're a student studying probability and statistics, a teacher preparing lessons, or a professional working with combinatorics, this calculator provides instant, accurate results with detailed step-by-step explanations.
Understanding when to use permutations versus combinations is crucial in mathematics, probability theory, and real-world applications. This calculator not only computes the results but also explains the difference, shows the complete solution process, and provides intelligent insights to deepen your understanding.
Understanding Permutations
What is a Permutation?
A permutation is an arrangement of items in a specific order. When calculating permutations, the order of selection matters, meaning "ABC" and "CBA" are considered different arrangements.
Permutation Formula
P(n, r) = n! / (n - r)!
- n = total number of items
- r = number of items to arrange
- n! = n factorial = n × (n-1) × (n-2) × ... × 2 × 1
Permutation Examples
Example 1: Race Results
How many ways can 3 medals (Gold, Silver, Bronze) be awarded to 8 runners?
P(8, 3) = 8! / (8-3)! = 8! / 5!
= (8 × 7 × 6 × 5!) / 5!
= 8 × 7 × 6 = 336 ways
Understanding Combinations
What is a Combination?
A combination is a selection of items where the order does NOT matter. When calculating combinations, {A, B, C} and {C, B, A} are considered the same selection.
Combination Formula
C(n, r) = n! / ((n - r)! × r!)
- n = total number of items
- r = number of items to choose
- C(n, r) is also called binomial coefficient, written as (n r)
Combination Examples
Example 2: Lottery
How many ways can you choose 6 numbers from 49 numbers (order doesn't matter)?
C(49, 6) = 49! / (43! × 6!)
= 13,983,816 ways
This is why lottery odds are so low!
Permutation vs Combination: Key Differences
| Aspect | Permutation | Combination |
|---|---|---|
| Order | Matters | Doesn't Matter |
| Formula | P(n,r) = n!/(n-r)! | C(n,r) = n!/((n-r)!×r!) |
| Result Size | Larger | Smaller (P(n,r) = C(n,r)×r!) |
| Example Use | Password, Race podium | Lottery, Team selection |
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Additional Resources
For more information about permutations, combinations, and combinatorics: