Permutations and Combinations Calculator
Calculate combinations nCr and permutations nPr with or without repetition, factorial expansions, and subset ordering.
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What Is a Permutations and Combinations Calculator?
A permutations and combinations calculator determines the number of distinct ways to choose and arrange r items from a collection of n total items. Combinations apply when selection order does not matter (e.g. lottery balls, poker hands), while permutations apply when order is critical (e.g. race rankings, security pin codes).
How to Use This Calculator
- Enter the total number of items in the population set (n).
- Enter the number of items to select in each subset (r).
- Compare combinations nCr (order does not matter) against permutations nPr (order matters).
- Review alternate counts with repetition allowed (n^r for permutations, C(n+r-1, r) for combinations).
Combinatorics Formulas
nPr = \frac{n!}{(n-r)!}, \quad nCr = \binom{n}{r} = \frac{n!}{r!(n-r)!}, \quad \text{Repetition: } n^r \text{ and } \binom{n+r-1}{r}Where n is total items, r is items chosen, and n! denotes the factorial product of all positive integers from 1 up to n.
Worked Example
Scenario: Find how many unique committees of 3 people can be selected from an 8-person team.
Set n = 8 (total candidates) and r = 3 (seats on committee).
Because committee roles are equal, order does not matter → use combinations nCr.
nCr = 8! / (3! × 5!) = (8 × 7 × 6) / (3 × 2 × 1) = 336 / 6 = 56.
If assigning distinct roles (President, VP, Treasurer), use permutations: nPr = 8! / 5! = 336.
Tips & Key Notes
- Remember: Permutation = Position matters; Combination = Clump / Collection where order is irrelevant.
- By definition, nC0 = nCn = 1, and nCr = nC(n-r).
- Factorial growth is extremely fast: 10! = 3,628,800, while 20! exceeds 2.43 quintillion.
Frequently Asked Questions
When should I use combinations instead of permutations?
Use combinations when rearranging the chosen items produces the exact same group (e.g., a hand of playing cards or a team). Use permutations when order creates a new outcome (e.g., a locker combination lock or running race results).
Why is a combination lock actually a permutation lock?
Because entering the numbers 12-34-56 in the order 56-12-34 will not open the lock! The order of entry is essential, making it mathematically a permutation.
What does repetition mean in combinatorics?
Repetition means an item can be selected more than once (e.g. creating a 4-digit PIN where digits 0000 or 1212 are allowed). Without repetition, each chosen item is removed from the remaining pool.
What is 0 factorial (0!) and why does it equal 1?
0! = 1 by mathematical convention and algebraic necessity. There is exactly one way to arrange zero objects: the empty set.
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