Permutation & Combination Calculator
Understanding Permutations and Combinations
Permutations and combinations are fundamental concepts in combinatorics, a branch of mathematics concerning the counting of finite discrete structures. Though they both involve selecting items from a larger set, they are distinguished by one critical factor: order.
What is a Permutation (nPr)?
A permutation refers to an arrangement of items where the order matters. If you are selecting a subset of items from a larger set and the sequence in which you arrange them is important, you are dealing with a permutation. Think of it as "arrangements."
The formula to calculate the number of permutations of 'r' items from a set of 'n' is:
nPr = n! / (n - r)!
Where 'n!' represents the factorial of n (the product of all positive integers up to n).
Real-World Example of a Permutation
Consider a race with 10 runners. In how many ways can the gold, silver, and bronze medals be awarded? Here, the order is crucial. Runner A coming first, B second, and C third is different from B first, A second, and C third.
- Total items (n) = 10
- Items to choose (r) = 3
- Calculation: 10P3 = 10! / (10 - 3)! = 10! / 7! = (10 × 9 × 8 × 7!) / 7! = 720.
There are 720 different ways to award the three medals.
What is a Combination (nCr)?
A combination refers to a selection of items where the order does not matter. When you are choosing a subset of items from a larger set and the sequence is irrelevant, you are dealing with a combination. Think of it as "selections" or "groups."
The formula for combinations is:
nCr = n! / [r! * (n - r)!]
This formula is similar to the permutation formula but includes an extra 'r!' in the denominator to account for the different arrangements of the chosen items, as they are all considered one combination.
Real-World Example of a Combination
Imagine you have a group of 10 friends, and you want to form a committee of 3 people. In this case, selecting friends A, B, and C for the committee is the same as selecting B, C, and A. The order doesn't change the composition of the committee.
- Total items (n) = 10
- Items to choose (r) = 3
- Calculation: 10C3 = 10! / [3! * (10 - 3)!] = 10! / (3! * 7!) = (10 × 9 × 8) / (3 × 2 × 1) = 120.
There are 120 different possible committees of 3 that can be formed.
Frequently Asked Questions (FAQ)
How do you calculate nPr?
To calculate permutations (nPr), you use the formula n! / (n - r)!. This means you find the factorial of the total number of items (n) and divide it by the factorial of the difference between the total items and the number of items being arranged (n - r).
How do you calculate nCr?
To calculate combinations (nCr), you use the formula n! / [r! * (n - r)!]. This is similar to the permutation formula, but you also divide by the factorial of the number of items being chosen (r!) to remove the duplicate arrangements.
When should you use permutation vs. combination?
The key question to ask is: "Does the order matter?"
- Use Permutations if order matters. Examples include arranging books on a shelf, creating passcodes, or determining finishing places in a competition.
- Use Combinations if order does not matter. Examples include picking lottery numbers, choosing players for a sports team, or selecting toppings for a pizza.
This calculator is for educational purposes only. Always double-check results for academic or professional use.
