Zero Factorial (0!)

The value of zero factorial is 1. Factorial of any number “n” is calculated by multiplying all the numbers between n and 1 (including n). So one might ask what is the value of zero factorial, the value of 0! factorial is 1 and this is calculated using various methods.

In this article we are going to learn about the definition of factorial, how factorial is calculated, the Derivation of 0! is equal to 1, Examples and FAQs related to Factorial, and others.

Definition of Factorial

Factorial of any number is calculated by finding the product of n and all numbers less than n, till it reaches 1. Another definition of factorial can be defined as the factorial of a whole number is the function that multiplies the number by every natural number less than it. The factorial of any number is represented by denoting an exclamation mark after it, symbolically it is written as n!. Factorials are used to calculate permutation and combination. Factorial of a number can be calculated in several ways. For example, if we have to calculate the factorial of 5, then it can be represented s 5! = 5 × 4 × 3 × 2 × 1 = 120. So, the value of 5! = 120.

Factorial of a Number Formulas

Below are the different formulas for calculating the factorial of a number

• n! = 1 × 2 × 3 × 4 × 5…………. × n
• n! = n × (n-1) × (n-2) × (n-3) × (n-4)………… × 1
• n! = n × (n-1)!
• n! = (n+1)!/(n+1)
• 0! = 1
• 1! = 1

How is Factorial Calculated?

Let us suppose we have to calculate the factorial of n, then the factorial of n is denoted by putting an exclamation mark after n i.e. n!

Value of n! = n × (n-1) × (n-2) × (n-3) × (n-4) × (n-5) …× 1

For example, If we have to calculate the value of 7!.

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1

7! = 5040

Another way of calculating factorial is by using the below formula for non-negative numbers. By taking 2 or three non-negative numbers, we can observe that the factorial of a number is calculated as,

n! = (n+1)!/(n+1)

What is Factorial of 0?

The value of Zero factorial is equal to 1. Symbolically, it can be represented as 0! = 1. We can prove that the value of zero factorial is equal to 1 in different ways. As factorial is used to calculate the permutation and combination of any number, logically the meaning of zero factorial is to arrange data that contains no value. So, the way of arranging any data which contains no values is in only one way. So the value of 0! is equal to 1.

0! = 1

Explain Zero Factorial

The value of 0! factorial is equal to 1. Let’s see how we can derive it using the above formula for calculating factorial:

We can write, n! = (n+1)!/(n+1)

0! = (0 + 1)!/(0 + 1)

⇒ 0! = (1)!/(1)

⇒ 0! = 1/1

⇒ 0! = 1

Thus, the factorial of 0 is one.

Derivation of Zero Factorial is Equal to 1

The formula for calculating the factorial of any number is equal to the product of all the positive numbers less than or equal to a number.

Formula for n! = n × (n-1) × (n-2) × (n-3) × (n-4)…….. × 1

Above formula can also be written as n! = n × (n-1)!

For the value of 1! = 1 × (1-1)!

1! = 1 × 0!

The value of LHS should always be equal to the value of RHS

For LHS = RHS, a value of 0! must be equal to 1

Hence, 0! = 1

Factorial of Negative Number

Factorial of a negative number is not defined/undefined. If we extend the definition of factorial using the gamma function then the factorial of a negative number is calculated, but in general, it is not defined. Let’s see how we can prove the factorial of negative numbers is undefined.

Formula for calculating the factorial of n! = (n+1)!/(n+1)

Calculating the value of (-1)! using the above formula: (-1)! = (-1+1)!/(-1+1)

(-1)! = (0)!/0

(-1)! = 1/0

Any value divided by 0 is undefined, so the negative number value is not defined.

Operations On Factorial

We perform basic mathematical operations such as addition, subtraction, multiplication, and division similar to we calculate for any number using factorial. Let’s understand this with examples,

5! + 4! + 0! = 120 + 24 + 1 = 145

⇒ 4! – 3! – 0! = 24 – 6 – 1 = 17

⇒ 5! × 0! = 120 × 1 = 120

⇒ 4! ÷ 0! = 4 ÷ 1 = 4

Read More,

• Permutations and Combinations
• Combinations

Sample Problems on Zero Factorial

Problem 1: Find the value of the given expression: 5! + 0! + 6! + 0! + 1!

Solution:

⇒ 5! + 0! + 6! + 0! + 1! = 5× 4×3 × 2×1 + 1 + 6×5×4 × 3×2×1 + 1 + 1 (We know that 0! = 1)

⇒ 5! + 0! + 6! + 0! + 1! = 120 + 1 + 720 + 1 + 1 = 843

Thus, 843 is the required answer.

Problem 2: Sim (5! + 0!) / (2! + 0!)

Solution:

= (120 + 1)/(2 + 1) (We know that 0! = 1)

= (121)/(3)

= 40.333

Practice Problems on Zero Factorial

Problem 1: Evaluate the expression: (3 + 2)!

Problem 2: Calculate the value of 5! / 5!

Problem 3: Find the value of (n + 1)! / n! for any positive integer n.

Problem 4: Determine the value of (2n)! / (n!)^2 for any positive integer n.

Problem 5: Compute the value of (4!)! / 4!.

Problem 6: If m! = 1, what is the possible value of m?

Problem 7: Find the value of (n – 1)! / n! for any positive integer n.

Problem 8: Simplify the expression: (0!)! + 1.

Problem 9: Evaluate the expression: (2!)! – 2!

Zero Factorial – FAQs

1. What is the Value of Zero Factorial?

The value of Zero factorial is equal to 1.

2. What is the Factorial of a Negative Number?

Factorial of negative number is undefined.

3. What are Factorial Formulas?

Some factorial formulas are,

• n! = 1 × 2 × 3 × 4 × 5…………. × n
• n! = n × (n-1) × (n-2) × (n-3) × (n-4)………… × 1
• n! = n × (n-1)!
• n! = (n+1)!/(n+1)

4. Why Does Zero Factorial Equal One?

For any number n! = n × (n – 1)

For 1

1! = 1 × (1 – 1)!

1! = 1 × 0!

For LHS and RHS to be equal , value of 0! should be equal to 1.

5. What is the Solution to 0 Factorial?

The value of Zero (0) factorial is 1.