Combinations and Permutations Calculator

Calculate permutations, arrangements, and combinations (simple and with repetition). Formula, result, and step-by-step explanation.

Combinatorics: Combinations, Arrangements, and Permutations

Combinatorics studies the ways in which elements can be selected and ordered from a set. This calculator computes permutations, arrangements, and combinations (simple and with repetition), showing the formula and result even for very large numbers.

The Six Operations of Combinatorics

Type	Order matters?	Repetition?	Formula
Simple permutations	Yes	No	P(n) = n!
Permutations with repetition	Yes	Yes	P'(n,k) = n^k
Simple arrangements	Yes	No	D(n,k) = n!/(n-k)!
Arrangements with repetition	Yes	Yes	D'(n,k) = n^k
Simple combinations	No	No	C(n,k) = n!/(k!(n-k)!)
Combinations with repetition	No	Yes	C'(n,k) = (n+k-1)!/(k!(n-1)!)

Practical Examples

Simple permutations: How many ways can 8 people sit at a table? P(8) = 8! = 40,320.

Simple arrangements: How many different podiums (gold, silver, bronze) are possible with 20 athletes? D(20,3) = 20x19x18 = 6,840.

Simple combinations: How many groups of 5 players can be formed from a roster of 25? C(25,5) = 53,130.

The Factorial

The factorial (n!) is the building block of combinatorics. It grows incredibly fast: 5! = 120, 10! = 3,628,800, 52! has 68 digits.

Frequently Asked Questions

What is the difference between combinations, arrangements, and permutations?▼

Permutations involve ordering all n elements. Arrangements choose k elements from n where order matters (e.g., a race podium). Combinations choose k elements from n where order does NOT matter (e.g., lottery draws). "With repetition" variants allow choosing the same element multiple times.

How is the factorial calculated?▼

The factorial of n (written n!) is the product of all integers from 1 to n. Example: 5! = 5x4x3x2x1 = 120. By convention, 0! = 1. The factorial grows very rapidly: 10! = 3,628,800, 20! exceeds 2 billion billion.

When should I use "with repetition" vs. "simple"?▼

Use "simple" (without repetition) when each element can be chosen at most once (e.g., drawing cards from a deck). Use "with repetition" when the same element can be chosen multiple times (e.g., a safe combination, PIN code, or choosing ice cream flavors with repeats).