Permutation and Combination Calculator

Calculate the number of arrangements and selections

Total Number of Items (n)

Number of Items to Choose (r)

Calculation Results

Permutations (nPr)

Order matters

Combinations (nCr)

Order does not matter

Formula Reference:
Permutation: $P(n, r) = n! / (n-r)!$
Combination: $C(n, r) = n! / (r! * (n-r)!)$

Tip: ‘n’ must be greater than or equal to ‘r’. Both ‘n’ and ‘r’ must be non-negative integers.

Permutations and Combinations: Understanding the Difference

The Permutation and Combination Calculator helps you determine the number of possible arrangements (permutations) and selections (combinations) you can make from a set of items. These concepts are fundamental in probability, statistics, and various fields of mathematics.

Key Distinction: The core difference between permutations and combinations lies in whether the order of selection matters. Permutations are about arrangements, while combinations are about selections.

What are Permutations?

A permutation is an arrangement of all or part of a set of objects, where the order of the objects matters. For example, if you have three items (A, B, C) and you want to arrange two of them, AB is different from BA.

The formula for permutations of ‘n’ items taken ‘r’ at a time is:

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

Where:

$n$ is the total number of items
$r$ is the number of items to choose and arrange
$!$ denotes the factorial (e.g., $5! = 5 \times 4 \times 3 \times 2 \times 1$)

What are Combinations?

A combination is a selection of all or part of a set of objects, where the order of selection does not matter. Using the same example, if you have three items (A, B, C) and you want to choose two of them, AB is considered the same as BA.

The formula for combinations of ‘n’ items taken ‘r’ at a time is:

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

Where:

$n$ is the total number of items
$r$ is the number of items to choose
$!$ denotes the factorial

How to Use the Permutation and Combination Calculator

Our Permutation and Combination Calculator is simple to use:

Enter ‘n’: Input the total number of items in your set.
Enter ‘r’: Input the number of items you wish to choose or arrange.
Click ‘Calculate’: The calculator will instantly display the number of permutations and combinations.
Reset: Clear the inputs to perform a new calculation.

Applications of Permutations and Combinations

These concepts are widely used in various fields:

Probability: Calculating the likelihood of events, such as drawing specific cards in poker.
Statistics: Sampling techniques and experimental design.
Computer Science: Algorithm design, especially in areas like cryptography and data arrangement.
Science & Engineering: Analyzing arrangements of molecules or circuit configurations.
Everyday Life: From determining password possibilities to figuring out how many ways a team can be selected.

Whether you’re solving a complex mathematical problem or just trying to understand the chances of winning the lottery, our Permutation and Combination Calculator is an invaluable tool.