Why 0!=1? |Why Zero Factorial (0!) is equal to one (1) ? | Concept Clarification | Factorial Concept

Why 0!=1? |Why Zero Factorial (0!) is equal to one (1) ? | Concept Clarification | Factorial Concept
Usually Students face difficulty to understand why Factorial of Zero is equal to One.With the help of this video, student will clarify their concept of Factorial and Clarify why 0! = 1
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I can understand why many of us have a hard time accepting the fact that the value of zero factorial is equal to one. It comes across as an absurd statement that there’s no way it can be true. We have a common perception of zero for being notorious because there’s something about it that can make any number associated with it either vanish or misbehave.
For instance, a large number such as 1,000 multiplied by zero becomes zero. It disappears! On the other hand, a nice number such as 5 divided by zero becomes undefined. It misbehaves. So it is okay to be skeptical why zero “suddenly” becomes one, a nice number, after treating it with some special operation.
Simple “Proof” Why Zero Factorial is Equal to One
Let katex is not defined be a whole number, where katex is not defined is defined as the product of all whole numbers less than katex is not defined and including katex is not defined itself.
What it means is that you first start writing the whole number katex is not defined then count down until you reach the whole number katex is not defined.
The general formula of factorial can be written in fully expanded form as
n! = n·(n-1)·(n-2)·...·3·2·1
or in partially expanded form as
n! = n · (n-1)!
A zero factorial is a mathematical expression for the number of ways to arrange a data set with no values in it, which equals one. In general, the factorial of a number is a shorthand way to write a multiplication expression wherein the number is multiplied by each number less than it but greater than zero. 4! = 24, for example, is the same as writing 4 x 3 x 2 x 1 = 24, but one uses an exclamation mark to the right of the factorial number (four) to express the same equation.
It is pretty clear from these examples how to calculate the factorial of any whole number greater than or equal to one, but why is the value of zero factorial one despite the mathematical rule that anything multiplied by zero is equal to zero?
The definition of the factorial states that 0! = 1. This typically confuses people the first time that they see this equation, but we will see in the below examples why this makes sense when you look at the definition, permutations of, and formulas for the zero factorial.
The Definition of a Zero Factorial
The first reason why zero factorial is equal to one is that this is what the definition says it should be, which is a mathematically correct explanation (if a somewhat unsatisfying one). Still, one must remember that the definition of a factorial is the product of all integers equal to or less in value to the original number-in other words, a factorial is the number of combinations possible with numbers less than or equal to that number.
Because zero has no numbers less than it but is still in and of itself a number, there is but one possible combination of how that data set can be arranged: it cannot. This still counts as a way of arranging it, so by definition, a zero factorial is equal to one, just as 1! is equal to one because there is only a single possible arrangement of this data set.
For a better understanding of how this makes sense mathematically, 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.
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