Permutation and Combination Calculator

In the study of combinatorics, we often need to determine the number of possible ways to select or arrange items from a larger set. Whether you are calculating the odds of winning a lottery, the number of possible PIN codes, or the arrangements of a committee, understanding permutations and combinations is essential.

This calculator allows you to compute both values quickly, supporting scenarios with or without repetition.

What is a Permutation?

A Permutation is an arrangement of items where the order matters. For example, if you are picking a first, second, and third-place winner in a race, the order (A-B-C) is different from (C-B-A).

The Permutation Formula

When choosing $r$ items from a set of $n$ without repetition:

$P(n, r) = \frac{n!}{(n - r)!}$

If repetition is allowed (like a digital lock where you can use the same number twice):

$P = n^r$

What is a Combination?

A Combination is a selection of items where the order does NOT matter. For example, if you are choosing 3 fruits to make a smoothie, choosing (Apple, Banana, Cherry) is the same as choosing (Cherry, Banana, Apple).

The Combination Formula

When choosing $r$ items from a set of $n$ without repetition:

$C(n, r) = \binom{n}{r} = \frac{n!}{r!(n - r)!}$

If repetition is allowed (also known as multiset or stars and bars selection):

$C(n, r) = \binom{n + r - 1}{r} = \frac{(n + r - 1)!}{r!(n - 1)!}$

How to Use This Calculator

Total Items (n): Enter the total number of items in the set.
Items Selected (r): Enter how many items you are picking from the set.
Type: Choose Permutation if the order of selection is important, or Combination if the order is irrelevant.
Repetition: Toggle this on if the same item can be selected more than once.

Key Differences At a Glance

Worked Examples

Example 1: Permutation (No Repetition)

How many ways can 3 people be seated in 5 available chairs?

$n = 5, r = 3$
Formula: $5! / (5-3)! = 120 / 2 = 60$
Result: 60 ways.

Example 2: Combination (No Repetition)

How many ways can a committee of 3 be chosen from 10 employees?

$n = 10, r = 3$
Formula: $10! / (3! * 7!) = (10 \times 9 \times 8) / (3 \times 2 \times 1) = 120$
Result: 120 ways.

FAQ

When should I use Permutation vs Combination?

Ask yourself: "Does the order of the items I pick change the outcome?" If yes (like a phone number), use Permutation. If no (like picking cards for a hand), use Combination.

Can r be larger than n?

In scenarios without repetition, $r$ cannot be larger than $n$ because you cannot pick more unique items than exist in the set. However, with repetition, $r$ can be any positive integer (e.g., picking 10 items from a set of 2 colors).

What does the '!' symbol mean?

It stands for Factorial. The factorial of a number is the product of all positive integers less than or equal to that number. For example, $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$ .

Why is 0! equal to 1?

In mathematics, $0!$ is defined as 1 to ensure that the formulas for permutations and combinations work consistently, specifically when $n = r$ .

What is a practical use for combinations with repetition?

This is often used in chemistry (counting ways to arrange atoms in a molecule) or in commerce (counting the number of ways to pick 5 sodas from 3 different brands).

Limitations and Precision

This calculator uses high-precision arithmetic (Decimal.js) to handle large factorials. However, for very large values of $n$ (e.g., $n > 1000$ ), the computation may become slow or result in numbers exceeding standard display limits. Results are provided in standard or scientific notation for readability.

Permutation And Combination