Factorial, Permutations (nPr), Combinations (nCr) Calculator

Mathematics frequently involves the counting and organizing of items, which is where factorials, permutations, and combinations are relevant. Whether you are a student, an educator, or someone working on probability challenges, our Factorial, Permutations & Combination Calculator allows you to quickly calculate results without the risk of manual calculation mistakes.

What is Factorial?

A factorial is a mathematical operation used to find the product of all positive integers up to a given number. It’s denoted by an exclamation mark (!).

Formula:

$$n!=n\times (n-1)\times (n-2)\times \dots \times 3\times 2\times 1$$

Example:

$$5!=5\times 4\times 3\times 2\times 1=120$$

So, 5 factorial equals 120.

What is a Permutation?

A permutation is an arrangement of objects in a specific order. The order of selection matters in permutations.

For instance, arranging the letters A, B, and C as ABC, ACB, BAC, etc., is all the different permutations.

Formula:

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

Where:
n = total number of objects
r = number of objects selected

Example:

If you have 5 books and want to arrange 3 of them on a shelf:

$$P(5,3)=\frac{5!}{(5-3)!}=\frac{120}{2}=60$$

So, there are 60 different ways to arrange 3 books out of 5.

Permutations with Repetitions:

Sometimes, repetition is allowed, meaning each item can be chosen more than once.

For example, forming 3-letter codes from the digits 0–9 (where digits can repeat).

Formula:

𝑛^𝑟

Where:
n = number of available items
r = number of positions

What is a Combination?

A combination refers to the selection of items without considering the order. In combinations, order doesn’t matter.

For example, selecting 3 fruits (apple, banana, orange) from a basket of 5 fruits. it doesn’t matter in which order you choose them.

Formula:

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

Where:
n = total number of objects
r = number of objects selected

Example:

If you have 5 students and want to form a team of 3:

$$C(5,3)=\frac{5!}{3!(5-3)!}=\frac{120}{6\times 2}$$

So, there are 10 ways to choose 3 students from 5.

Combination with Repetitions:

In some cases, repetition is allowed, such as choosing scoops of ice cream where you can select the same flavor multiple times.

Formula:

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

Where:
n = total number of available types
r = number of selections

How to Use Our Factorial, Permutations & Combination Calculator

1. Select the operation you want to perform: Factorial, Permutation, or Combination.
2. Enter the values of n (total items) and r (items chosen).
3. The calculator displays: Factorial result (if chosen), Permutation, and Combination result