Permutation and Combination Calculator
Permutations, nPr = n! / (n - r)!
Combinations, nCr = n! / (r!(n - r)!)
Example:
5P3 = 5! / (5-3)! = 5 × 4 × 3 = 60
5C3 = 5! / (3!(5-3)!) = (5 × 4) / (2 × 1) = 10
Permutation and Combination
Permutations and combinations are important concepts in combinatorics, the branch of mathematics that deals with counting and arranging objects in a set. The main difference lies in the importance of order:
A permutation considers order important.
A combination ignores order.
For example, in a combination lock, entering the code 1-2-9 is not the same as 2-9-1, which makes it a permutation, not a true combination in mathematical terms. This calculator focuses on selections without replacement, meaning once an item is chosen, it cannot be selected again. It does not support repeated values such as 3-3-3.
What Are Permutations?
Permutations represent the number of ways to arrange a smaller group (r) of items selected from a larger set (n), where the order matters.
Permutation formula (without replacement):
nPr = n! / (n – r)!
This is often called a partial permutation.
Example: Selecting a Captain and Goalkeeper
Suppose you have a soccer team of 11 players, and you want to choose a captain and a goalkeeper:
Choose the captain: 11 options
Choose the goalkeeper: 10 options (since the captain can’t be selected again)
Total permutations = 11 × 10 = 110
Using the formula:
11P2 = 11! / (11 – 2)! = 11! / 9! = 11 × 10 = 110
This is a case of permutations without repetition, since the same player can’t be both captain and goalkeeper.
Note on Permutations With Repetition
Although not supported in this tool, the formula for permutations with repetition is:
n^r
What Are Combinations?
Combinations refer to selections where the order doesn’t matter. Choosing A and B is the same as choosing B and A.
Combination formula (without replacement):
nCr = n! / [r! × (n – r)!]
This formula helps you find how many unique groups can be formed from a set.
Example: Choosing 2 Strikers from 11 Players
Let’s say you want to select 2 strikers. Since both players will have the same role, the selection order doesn’t matter:
Total permutations: 11 × 10 = 110
Remove duplicates caused by order: divide by 2! (which is 2)
11C2 = 11! / [2! × 9!] = (11 × 10) / 2 = 55 combinations
Combinations remove redundant arrangements since order is not important.
Note on Combinations With Repetition
This calculator doesn’t compute combinations with repetition, but here’s the formula:
nCr = (n + r – 1)! / [r! × (n – 1)!]
Summary Table
| Feature | Permutations | Combinations |
|---|---|---|
| Order Matters? | Yes | No |
| Formula | nPr = n! / (n – r)! | nCr = n! / [r! × (n – r)!] |
| With Replacement | Not supported | Not supported |
| Use Case | Arranging positions or rankings | Forming groups or teams |
Whether you’re solving math problems or exploring statistics, this calculator simplifies how many ways you can arrange or select items from a group. Just input your values for n and r to get instant results!