Permutation and Combination Calculator
What are Permutations and Combinations?
Permutations and combinations are mathematical concepts that help us count the number of possible arrangements or selections from a set of items.
Permutations (nPr)
A permutation is an arrangement of objects where the order matters. For example, the permutations of the letters A, B, C are: ABC, ACB, BAC, BCA, CAB, CBA.
The formula for calculating the number of permutations of r items selected from a set of n items is:
P(n,r) = n! / (n-r)!
Combinations (nCr)
A combination is a selection of objects where the order does not matter. For example, the combinations of 2 letters from A, B, C are: AB, AC, BC.
The formula for calculating the number of combinations of r items selected from a set of n items is:
C(n,r) = n! / (r! × (n-r)!)
Applications of Permutations and Combinations
- Probability and statistics
- Cryptography and security
- Game theory and puzzles
- Sports scheduling
- Genetic sequencing
Examples
Permutation Example
How many different ways can 5 runners finish in a race?
This is a permutation of 5 items taken 5 at a time: P(5,5) = 5! = 5 × 4 × 3 × 2 × 1 = 120 different possible race outcomes.
Combination Example
How many different committees of 3 people can be formed from a group of 8 people?
This is a combination of 8 items taken 3 at a time: C(8,3) = 8! / (3! × 5!) = 56 different possible committees.