Mostly it is based on convention, when one wants to define the quantity for example. An intuitive way to look at it is counts the number of ways to arrange distinct objects in a line, and there is only one way to arrange nothing.

Yes, precisely there is a unique function (with empty graph), which happens to be a bijection (). Note, that is the number of bijections .

We need $0!$ to be defined as $1$ so that many mathematical formulae work. For example we would like $$n! = n \times (n-1)!$$ to work when $n=1,$ ie $1! = 1 \times 0!.$ Also we require that the formula for the number of ways of choosing $k$ objects from $n$ is valid for $k=n.$ ie $${n \choose k} = \frac{n!}{k!(n-k)!}$$ is valid when $k=n.$

Things need to work when we extend our definition of the factorial via the gamma function.

$$\Gamma(z) = \int\limits_0^\infty t^{z-1} e^{-t} \,\mathrm{d}t,\qquad \Re(z)>0.$$

The above gives $\Gamma(n)=(n-1)!$ and so we require $0!=1,$ since $\Gamma(1)=1.$

Hint: In mathematics, zero factorial is the expression that means to arrange the data containing no values. The value of n! is given by $n!=n\\times \\left( n-1 \\right)\\times \\left( n-2 \\right)\\times \\ldots \\ldots \\ldots \\times 1$ . The given equation can also be written as $n!=n\\times \\left( n-1 \\right)!$ . 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 $n!=n\\times \\left( n-1 \\right)\\times \\left( n-2 \\right)\\times \\ldots \\ldots \\ldots \\times 1$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 $n!=n\\times \\left( n-1 \\right)\\times \\left( n-2 \\right)\\times \\ldots \\ldots \\ldots \\times 1$The value of n! from the above can be also written as $n\\times \\left( n-1 \\right)!$$\\Rightarrow n!=n\\times \\left( n-1 \\right)!$Considering the value of n equal to 1,$\\Rightarrow 1!=1!\\times \\left( 1-1 \\right)!$$\\Rightarrow 1!=1!\\times \\left( 0 \\right)!$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.

Because the only way to arrange 0 distinct objects is one way.

Because a/b = x which means a = bx. For example : 6/2 = 3 which is same as saying 6 = 2*3. So when we divide 1 (or any number) by 0, it implies that 1/0 = x i.e. 1 = 0*x , which we know is not possible as anything multiplied by 0 gives you 0. Therefore we say it is undefined and not infinity.