Permutation & Combination Calculator

How to Use the Permutation & Combination Calculator

Enter the total number of items (n) and the number of items being chosen or arranged (r). The calculator instantly displays P(n,r) for permutations where order matters and C(n,r) for combinations where it does not. It also shows the intermediate factorial values and the full formulas so you can follow the computation step by step.

Counting problems appear throughout probability, statistics, and discrete mathematics. Whether you are calculating lottery odds, planning tournament brackets, or working through a textbook exercise, this tool removes the tedious arithmetic and lets you focus on understanding the concepts.

Permutations Explained

A permutation counts the number of ordered arrangements of r items from a set of n. The formula is P(n,r) = n! / (n−r)!. For example, the number of ways to arrange 3 books out of 10 on a shelf is P(10,3) = 10! / 7! = 720. Order matters because putting Book A first and Book B second is different from Book B first and Book A second.

Combinations Explained

A combination counts selections where order does not matter. The formula is C(n,r) = n! / (r! × (n−r)!). Choosing a 3-person committee from 10 people gives C(10,3) = 120. Since the committee {Alice, Bob, Carol} is the same as {Carol, Alice, Bob}, we divide by the number of ways to arrange the chosen items (r!).

Understanding Factorials

The factorial function n! equals 1 × 2 × 3 × ... × n. It counts the number of ways to arrange n distinct items. Factorials grow extremely fast: 10! = 3,628,800 and 20! exceeds 2.4 quintillion. By convention, 0! = 1, which ensures the formulas work correctly when r = 0 or r = n.

Real-World Applications

Lottery odds use combinations because the order in which numbers are drawn does not matter. Password complexity analysis uses permutations because character order matters. Tournament seedings, DNA sequence analysis, poker hand rankings, and cryptographic key spaces all rely on counting principles.

Frequently Asked Questions

What is the difference between a permutation and a combination?

Permutations count ordered arrangements; combinations count unordered selections. P(n,r) is always greater than or equal to C(n,r).

What is the formula for permutations?

P(n,r) = n! / (n−r)!, where n is the total items and r is the number chosen.

What is the formula for combinations?

C(n,r) = n! / (r! × (n−r)!), also written as “n choose r”.

What is a factorial?

n! is the product of all positive integers up to n. For example, 5! = 120. By definition, 0! = 1.

When do you use permutations vs combinations?

Use permutations when order matters (passwords, rankings). Use combinations when order does not matter (committees, lottery picks).