What is the difference between a permutation and a combination?

The key difference is order. In a permutation, the order of the selected items matters (e.g., a lock combination like '1-2-3' is different from '3-2-1'). In a combination, the order does not matter (e.g., a team of 'Ann, Bob, Chris' is the same as 'Chris, Bob, Ann').

What is a factorial (n!)?

A factorial, denoted by 'n!', is the product of all positive integers up to that number. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are fundamental to calculating both permutations and combinations.

When would I use a permutation?

Use permutations when you are arranging items in a specific order, such as arranging books on a shelf, assigning roles in a play, or creating a password from a set of characters.

Permutation and Combination Calculator

How Does This Calculator Work?

This calculator determines the number of ways you can arrange or select items from a set, based on two key concepts in combinatorics:

Permutation (nPr): A permutation finds the number of ways to arrange 'r' items from a set of 'n' items where the order matters. For example, arranging 3 people in 3 chairs.
Formula: nPr = n! / (n - r)!
Combination (nCr): A combination finds the number of ways to choose 'r' items from a set of 'n' items where the order does not matter. For example, choosing a committee of 3 people from a group of 10.
Formula: nCr = n! / (r! * (n - r)!)

Both formulas rely on factorials (e.g., 5! = 5 × 4 × 3 × 2 × 1).

The Surprising History of Combinatorics

The study of permutations and combinations—the heart of combinatorics—has ancient roots. Evidence suggests that early forms of this math were explored in India as far back as the 6th century BC. The ancient Indian text, the *Bhagavati Sutra*, contains the first known mention of the combination formula (nCr).

Later, Indian mathematicians like Mahavira in the 9th century and Bhaskara II in the 12th century developed these ideas further. However, combinatorics truly flourished in Europe during the 17th century, driven by the same force that fueled probability theory: gambling. Mathematicians like Blaise Pascal and Pierre de Fermat analyzed games of chance, which required them to systematically count the number of possible outcomes, leading to the formalization of the nPr and nCr formulas we use today.

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While you're here, check out some of our other popular math and statistics calculators:

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Frequently Asked Questions (FAQ)

What is the main difference between permutation and combination?

The single most important difference is order. If the order of the items matters, use a permutation. If the order doesn't matter, use a combination. For example, picking a president, vice president, and treasurer is a permutation because the roles are different. Picking a 3-person committee is a combination because all members have the same role.

Why are there always more permutations than combinations?

For any given n and r, there will always be more (or equal) permutations than combinations. This is because permutations count every different ordering of a group as a separate possibility, while combinations count each group only once. The combination formula essentially divides out all the duplicate arrangements.

What does n! (factorial) mean?

The factorial of a non-negative integer 'n', denoted by n!, is the product of all positive integers less than or equal to n. For example, 4! = 4 × 3 × 2 × 1 = 24. By convention, the value of 0! is 1.