Factorial Calculator

Factorial Calculator:

This factorial calculator finds the factorial (n!) of any non-negative integer and shows the complete step-by-step expansion. It also lets you add, subtract, multiply, and divide factorials of two different numbers.

What is a Factorial?

In mathematics, the factorial function (n!) is the product of every positive number from n down to 1. It shows he number of ways to arrange or order objects.

For example:

• If n = 5, then 5! is n ! = 5 × 4 × 3 × 2 × 1 = 120
• If n = 7, then 7! is 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040

Factorial Formula:

n! = n × (n-1) × (n-2) × ……. × 1

Where,

• n is the positive integer for which you want to calculate the factorial.

How to Use This Factorial Calculator?

Follow these steps:

1. Choose the Operation: Select whether you want to calculate a single factorial or perform addition, subtraction, multiplication, or division between two factorials
2. Enter the Non-Negative Integers: Input the number for a single factorial or two numbers if operating between factorials
3. Click “Calculate”: Press the calculate button and view the results

What You will See:

• Final numeric result 
• Step-by-step factorial expansion 
• For operations, it also shows the factorial expansion and the steps of the operation performed 

Supported Operations on Factorials:

Simple Factorial:

Example:

5! = 5 × 4 × 3 × 2 × 1 = 120

Add Two Factorials:

Example:

5! = 5 × 4 × 3 × 2 × 1 = 120

4! = 4 × 3 × 2 × 1 = 24 

5! + 4! = 120 + 24 = 144

Subtract Two Factorials:

Example:

6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

5! = 5 × 4 × 3 × 2 × 1 = 120 

6! - 5! = 720 - 120 = 600

Multiply Two Factorials:

Example:

4! = 4 × 3 × 2 × 1 = 24

3! = 3 × 2 × 1 = 6

4! × 3! = 24 × 6 = 144

Divide Two Factorials:

Example:

6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

4! = 4 × 3 × 2 × 1 = 24

6! ÷ 4! = 720 ÷ 24 = 30

Why is it Not Possible to have a Negative Factorial?

The factorial formula clearly shows that it could only apply to the positive numbers, which prevents going below 0. As it gives the number of ways to arrange or permute objects, you can not arrange a negative number of objects. Therefore, the factorial is undefined for negative numbers, and it's not possible to have a negative factorial.

Mathematically, the factorial formulas contain a series of numbers multiplied in a descending order, which stops at 1 (or 0! = 1). It means there is no valid result for negative numbers. For this reason, our factorial expression calculator is designed to compute only the valid inputs where n ≥ 0.

What is the Factorial of 0?

The Factorial of Zero (0!) is a Special Case:

The factorial of zero is equal to one (0! = 1). It appears to be a mistake, but it is a fact. It's an important rule based on two mathematical concepts, combinatorics and the recursive definition of the factorial.

1. Combinatorial Reason (The Arrangement Rule):

The factorial shows the possible number of arrangements that you can use to arrange objects (a permutation).

For Example:

If you have 4 objects, there are 4! = 24 ways to arrange them.

If you have zero objects, then it means there is only one way to arrange the objects. Therefore, 0! must be equal to 1.

2. The Recursive Formula Reason (The Logic Must Continue):

n! = n × (n-1)!

(0!) = (0! x (0-1)!)

We know that the factorial of (n) is only defined when (n>0), so that’s where the confusion takes place. When the term ((0-1)! = (-1)!) is computed, the factorial becomes undefined because the factorials are not defined for negative numbers. 

The problem is not that you cannot calculate it, but simply that the factorial represents the ways to arrange objects, and you can not arrange a negative number of objects. If we place the value of (0!) to (1), we can get the expected values for (n!). Our factorials calculator also follows the standard definition and determines the factorial of zero as 1, along with other positive integers as well.

How to Calculate Factorial of a Number?

Let’s see some examples to understand the complete step-by-step calculations:

Example (Simple Factorial):

Calculate the factorial of 8.

Solution:

Here, n = 8

Step 1:

8! = 8 × (8-1) × (8-2) × (8-3) × (8-4) × (8-5) × (8-6) × (8-7)\)

Step 2:

8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 ×1

Step 3:

8! = 40320

To calculate (n! + m!):

Example:

Add the factorials of 3 and 4.

Solution:

Here,

• n = 3
• m = 4

Step 1:

Find n! = 3 × (3-1) × (3-2) = 3 × 2 × 1

Step 2:

Find m! = 4 × (4-1) × (4-2) × (4-3) = 4 × 3 × 2 × 1 = 24

Step 3:

n! + m! = 6 + 24 = 30

To calculate (n! - m!):

Example:

Subtract the factorials of 5 and 3.

Solution:

Here,

• n = 5 
• m = 3

Step 1:

Find n! = 5 × (5-1) × (5-2) × (5-3) × (5-4) = 5 × 4 × 3 × 2 × 1 = 120

Step 2:

Find m! = 3 × (3-1) × (3-2) = 3 × 2 × 1 = 6

Step 3:

n! - m! = 120 - 6 

n! - m! = 114

To calculate (n! x m!):

Example:

Multiply the factorial of 7 and 4.

Solution:

Here, 

• n = 7 
• m = 4

Step 1:

Find n! = 7 × (7-1) × (7-2) × (7-3) × (7-4) × (7-5) × (7-6) 

7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040

Step 2:

Find m! = 4 × (4-1) × (4-2) × (4-3)

 4 × 3 × 2 × 1 = 24

Step 3:

n! × m! = 5040 × 24 = 120960

To calculate (n! / m!):

Example:

Divide the factorial of 5 and 6.

Solution:

Here,

• n = 5 
• m = 6

Step 1:

Find 5! = 5 × (5-1) × (5-2) × (5-3) × (5-4) 

5 × 4 × 3 × 2 × 1 =120

Step 2:

Find m! = 6 × (6-1) × (6-2) × (6-3) × (6-4) × (6-5)

6 × 5 × 4 × 3 × 2 × 1 = 720

Step 3:

n!/m! = 120/720 = 0.16666

Rather than going through slow manual calculation, an online factorial calculator provides a quick and reliable way to evaluate n! for any non-negative number.

FAQ’s:

What does the n! Mean?

The factorial of a number n means to multiply this number by each positive integer below it until you reach 1.

For example:

5! = 5 × 4 × 3 × 2 × 1 = 120

Where are Factorials Used?

Factorials are commonly used in counting problems, including combinations and permutations. If you are dealing with a large data set and need results quickly, then using an online combination calculator easily finds the possible number of selections for you. And, if your concern is to determine the number of possible arrangements of items, then using a permutation calculator is the ideal option.

How to calculate factorial in Excel?

To calculate the factorial in Excel, use the function (=FACT).

What is N factorial times n factorial?

If you multiply a factorial by itself, then the expression becomes: 

n! × n! = (n!)²

Example:

3! × 3! = 6 × 6 = 36

Basic Values of Factorials:

Let's check out the given table.

Factorial	Answer
n!	n(n-1)...1
0!	1
1!	1
2!	2
3!	6
4!	24
5!	120
6!	720
7!	5040
8!	40320
9!	362880
10!	3628800
12!	479001600
15!	1307674368000
20!	2432902008176640000
45!	1.1962222086548E+56
50!	3.0414093201713E+64
100!	9.3326215443944E+157

Final Words:

Factorials are largely used in mathematics and statistics, where they help in calculating the combinations and permutations, and are crucial in calculus, including the binomial theorem, Taylor Series, and derivatives. By using n factorial calculator, students, teachers, pr professionals can easily find the factorial of any non-negative integer and can perform different arithmetical operations on them, making complex calculations easier and faster. 

If you are exploring number properties beyond factorials, you may also find a prime factor calculator helpful. It breaks any integer into its prime factors, which is useful in simplifying expressions, understanding divisibility, and solving certain number theory problems.

References:

1. From the source of Wikipedia: Factorial, Definition, Rate of growth, and approximations for large n, and much more! 
2. The source of Khan Academy: n! function (all you need to know about it).
3. The xaktly site provides: The role of factorials in math (ultimate guide).