Question

How can the factorial of $0$ be $1$?

Answer 1

We can define a zero factorial as a mathematical expression for the number of ways in which we can arrange a set of data with no values in it, which equals one. In general, we define the factorial of a number as a shorthand way to write an expression of multiplication, in which the number is multiplied by each number less than it but greater than or equal to $1$. For example, we can write $5!=120$ as $5 \times 4 \times 3 \times \2 \times 1$ but we use an exclamation mark to the right of the factorial number (five) to express the factorial of the number.

Complete step by step solution :
We can show this by simply the definition of factorial which states that the factorial of $0$ is $1$. Although, there are many other ways to prove this statement.
Let $n$ be a whole number
Then $n!$ is defined as the product of integers including $n$ itself and everything below it.
It means is that we will first start writing the whole number n then count down until you reach the whole number $1$.
The general formula of factorial can be written in fully expanded form as
$n!\text{ }=\text{ }n\cdot \left( n-1 \right)\cdot \left( n-2 \right)\cdot ...\cdot 3\cdot 2\cdot 1$
or in partially expanded form as
$n!\text{ }=\text{ }n\text{ }\cdot \text{ }\left( n-1 \right)!$
We know with absolute certainty that$1!=1$, where$n\text{ }=\text{ }1$.
If we substitute that value of $n$ into the second formula which is the partially expanded form of $n!$, we obtain the following:

n!=n\cdot \left( n-1 \right)! \\\ 1!=1\text{ }\cdot \text{ }\left( 1-1 \right)! \\\ 1!=1\text{ }\cdot \text{ }\left( 0 \right)! \\\ 1!\text{ }=\text{ }1\text{ }\cdot \text{ }0! \\\ \end{array}$$ If we assume $$0!$$ to be not equal to $$1$$ , then $$1!$$ will not be given to $$1$$ , which is a contradiction to the fact that $$1!=1$$ and therefore 0! must be equal to $$1$$ . **Note:** A factorial is the number of combinations possible with numbers less than or equal to that number. It's important to note that factorials like these are used to determine possible orders of information in a sequence, also known as permutations, which can be useful in understanding that even though there are no values in an empty or zero set, there is still one way that set is arranged.