Others have explained the 0! = "number of ways to rearrange 0 people" part, so I'll try to convince you that 0! shouldn't be 0. You can think of 2 lines of people, with 2 distinct people in first one and 3 in the second. How many distinct way can you rearrange them, assuming that no one leaves their line? Well, as for the first line, there are 2! arrangements. As for the second one, the answer is 3!. Then, since you can pair 1 way of arranging the first line with 1 way of arranging the second line to make 1 distinct arrangement of the 2 lines, the number of distinct arrangements is 2! × 3!. Now, what if the first line is empty? Then the answer should be 0! × 3!. If 0! = 0, this number should be 0. But you should get 3! different arrangements when shuffling the second line, no? Then adding an empty first line shouldn't suddenly make the number of arrangement 0? Look just how convenient it is for the number of permutation of an empty set to be 1!

Hint: In mathematics, zero factorial is the expression that means to arrange the data containing no values. The value of n! is given by . The given equation can also be written as . we need to substitute n=1 to get the value of 0!Complete step by step answer:Factorial of a number in mathematics is the product of all the positive numbers less than or equal to a number. The multiplication happens to a given number down to the number one or till the number one is reached.Example: Factorial of n is n! and the value of n! is Definition 1:In mathematics, zero factorial is the expression that means to arrange the data containing no values. Factorial is used to define possible data sets in a sequence also known as permutation. Order is important in the case of permutations. As per the same, if there are no values like in an empty or zero set there is still a single arrangement possible.As there is no data to arrange, the value becomes eventually equal to one.Definition 2:Combinations usually are the number of ways the objects can be selected without replacement.Order is not usually a constraint in combinations, unlike permutations.Factorial of a number in mathematics is the product of all the positive numbers less than or equal to a number. But there are no positive values less than zero so the data set cannot be arranged which counts as the possible combination of how data can be arranged (it cannot).Thus, 0! = 1.Definition 3:Factorial of a number in mathematics is the product of all the positive numbers less than or equal to a number. Example: Factorial of n is n! and the value of n! is The value of n! from the above can be also written as Considering the value of n equal to 1,The value of LHS should be equal to RHS as 1! is always equal to 1!For the above condition to be true, The value of 0! must be equal to 1.The value of 0! =1.Note: The factorial of a number is denoted by an exclamation mark. Factorial of a number only deals with natural numbers so zero is omitted. The multiplication of any factorial takes place down to 1 and not zero. Factorials are usually used in the context of solving permutations and combinations.

Yes, precisely there is a unique function $\emptyset \to \emptyset$ (with empty graph), which happens to be a bijection ($\operatorname{id}_\emptyset$). Note, that $n!$ is the number of bijections $\{1,\dots, n\}\to \{1,\dots,n\}$.

One definition of the factorial that is more general than the usual

$$ N! = N\cdot(N-1) \dots 1 $$

is via the gamma function, where

$$ \Gamma(N) = (N-1)! = \int_0^{\infty} x^{N-1}e^{-x} dx $$

This definition is not limited to positive integers, and in fact can be taken as the definition of the factorial for non-integers. With this definition, you can quite clearly see that

$$ 0! = \Gamma(1) = \int_0^{\infty} e^{-x} dx = 1 $$

If you are starting from the "usual" definition of the factorial, in my opinion it is best to take the statement $0! = 1$ as a part of the definition of the factorial function, as anything else would require proofs using the factorial to include special cases for $0!$ and $1!$. It's a definition that is consistent and makes our lives easier.