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What is the symbol of factorial?
What is the meaning of 5 factorial?
What is the value of 7!?
. 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. You don't calculate $0.5!$ by hand. Decimal factorials do not make sense. The Gamma function might do what you want, but it's not the easiest to do by hand as it's a pretty unfriendly integral, at least in general. You might be able to calculate $\Gamma(1.5)$ (which would corespond to $0.5!$) by hand if you really wanted to: $$ \Gamma(1.5) = \int_0^\infty \sqrt x e^{-x} dx $$ and it turns out to be $\frac12\sqrt\pi$.
You can use Stirling's approximation, but it won't work well for small numbers like $\frac12$. For larger numbers, Stirling's approximation is quite good whether for integers or non-integers. It says: $$n! \approx \sqrt{2\pi n}\Bigl(\frac ne\Bigr)^n.$$
For example, $12.7!\approx 2.8616\times10^9$ and the Stirling formula gives around $2.843\times10^9$.
Of course, how to evaluate Stirling's formula with pencil and paper is an interesting question all by itself.
For smaller numbers of the form $n+\frac12$ your best bet is just to know that $$\frac12! = \frac12\sqrt\pi$$ and then use the rule that $(x+1)! = (x+1)x!$ so for example $\frac32! = \frac32\cdot \frac12! = \frac34\sqrt\pi$.
For numbers that are not half-integers, I have no good suggestions. There may be something involving the reciprocal factorial function (that is, $x\mapsto \frac1{x!}$) that can be calculated with some accuracy. There is a MacLaurin series that will be accurate, but it does not look easy to calculate with pen and paper.
I saw a post on this sub a few days ago (sorry, I tried to find it with no luck) where someone pointed out that: 1*2*3*4*5*6*7*8*9*10 = 1*2*3*4*5*6*7*6*5*4*3*2*1 It makes sense because the first 1 through 7 cancel out, then the prime factorization of 8*9*10 (2⁴ * 3²\ * 5) matches that for 1-6.
My question is: are there other examples of this? To formalize, for any two real number m and n, such that m<n, how many examples are there where n! = m! * (m+1)! (The above example of course being m=6 and n=10)
My gut intuition is that it would have to be during a decent length string of composite numbers.