Permutation Calculator
This permutation calculator computes the number of possible arrangements (permutations) and selections (combinations) of a set of items. Enter your values for n (total items) and r (selected items) to calculate permutations with and without repetition.
Results
| Permutation Type | - |
|---|---|
| Total Items (n) | 0 |
| Selected Items (r) | 0 |
| Number of Arrangements | 0 |
| Formula | - |
Permutation Types
| Type | Repetition | Order Matters | Formula |
|---|---|---|---|
| Permutation | No | Yes | n! / (n-r)! |
| Permutation with Repetition | Yes | Yes | n^r |
| Combination | No | No | n! / (r!(n-r)!) |
| Combination with Repetition | Yes | No | (n+r-1)! / (r!(n-1)!) |
About Permutations and Combinations
Permutations and combinations are fundamental concepts in probability and statistics that deal with counting possible arrangements of items. While they may seem similar, they have important differences based on whether order matters and whether repetition is allowed.
Permutations (Order Matters)
A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement. For example, the permutations of the letters A, B, and C would be ABC, ACB, BAC, BCA, CAB, and CBA.
The number of permutations of n objects taken r at a time is given by the formula:
P(n, r) = n! / (n - r)!
Permutations with Repetition
When repetition is allowed, the number of possible permutations increases. For example, if you're creating a 3-digit code where each digit can be 0-9, you have 10 options for each digit.
The formula for permutations with repetition is:
P(n, r) = n^r
Combinations (Order Doesn't Matter)
A combination is a selection of items from a larger pool where order doesn't matter. For example, the combination of letters A, B, and C taken 2 at a time would be AB, AC, and BC (but BA, CA, and CB are considered the same as AB, AC, and BC respectively).
The number of combinations is given by:
C(n, r) = n! / (r!(n - r)!)
Combinations with Repetition
When repetition is allowed in combinations, the number increases. This is common in problems where you're selecting items that can be repeated, like choosing flavors where you can have multiple scoops of the same flavor.
The formula for combinations with repetition is:
C(n, r) = (n + r - 1)! / (r!(n - 1)!)
Practical Applications
Permutations and combinations have wide applications in:
- Probability calculations
- Statistics and data analysis
- Computer science algorithms
- Cryptography
- Game theory
- Genetics and biology