Solveeit Logo
Login

Question

Question: What is a factorial of \(0?\)...

What is a factorial of 0?0?

Explanation

Solution

The factorial of a natural number nn is the product of all the natural numbers less than or equal to n.n. That is, 1.2.3...(n−1)n.1.2.3...\left( n-1 \right)n. The product of all the natural numbers from11 to (n−1)\left( n-1 \right) is the factorial of (n−1).\left( n-1 \right). That is, 1.2.3...(n−2)(n−1).1.2.3...\left( n-2 \right)\left( n-1 \right).

Complete step by step solution:
The factorial of a natural number nn is defined as n!=1.2.3...(n−1)n.n!=1.2.3...\left( n-1 \right)n.
The symbol !! is used to denote the factorial. And we call n!n! as nn factorial.
As defined above, the factorial of nn is the product of the natural numbers less than or equal to n.n.
According to our definition,
The factorial of 1,1, that is, 1!=1.1!=1.
The factorial of 2,2, that is, 2!=1×2=2.2!=1\times 2=2.
The factorial of 3,3, that is 3!=1×2×3=6.3!=1\times 2\times 3=6.
The factorial of 4,4, that is 4!=1×2×3×4=24.4!=1\times 2\times 3\times 4=24.
Similarly, we can find the factorial of any natural number.
Consider 2!=1×2.2!=1\times 2.
We have 1!=1.1!=1.
Put this value in the factorial of 2.2.
We will get 2!=1!×2.2!=1!\times 2.
Take 3!=1×2×3.3!=1\times 2\times 3.
Now we have 2!=1×2.2!=1\times 2.
Let us substitute this in 3!.3!.
We will get 3!=2!×3.3!=2!\times 3.
Also, consider the factorial of 4.4.
It is found that 4!=1×2×3×4.4!=1\times 2\times 3\times 4.
We have already obtained that 1×2×3=3!1\times 2\times 3=3!
When we substitute this in the value of 4!,4!, we will get 4!=3!×4.4!=3!\times 4.
From this we can discover the fact about the factorial which says that the factorial of a natural number nn is a product of all the natural numbers less than or equal to itself and thus, the factorial of a natural number nn is a product of the factorial of the natural number (n−1),\left( n-1 \right), which is less than nn and greater than every other natural numbers less than n,n, and nn itself.
Therefore, we can write, n!=1.2.3....(n−1)n=(n−1)!nn!=1.2.3....\left( n-1 \right)n=\left( n-1 \right)!n
Also, we can write n!=(n−2)!(n−1)n.n!=\left( n-2 \right)!\left( n-1 \right)n.
This is how we define the factorial of a natural number.
But we have an already defined fact that the factorial of 00 is 1.1.
That is, 0!=1.0!=1.
So, unlike any other cases, 00 can be put in the denominator of a fraction.
For example, 1=10!.1=\dfrac{1}{0!}.

Note: If we just use the fact n!=(n−1)!n,n!=\left( n-1 \right)!n, we can write 1!=0!×1.1!=0!\times 1.
We know that any number multiplied with 11 gives that number itself.
Also, 1!=1.1!=1.
Therefore, 0!=1.0!=1.
Although this is not an established way to say that 0!=1,0!=1, we often use this method.