What is factorial (n!)
When you saw an exclamation point in maths for the first time, you probably got shocked or even thought that there was some kind of mistake or typo. But the reality is different: this exclamation point in maths is called the factorial or n-factorial. The factorial is a reasonably unknown operator that can, in fact, be viewed more as an abbreviation than an actual operator, at least at the beginning.
It is important not to confuse the factorial with the prime factorization of a number, which is a way of obtaining the prime numbers that, when multiplied together, give your number. Prime factorization has its uses in maths and is arguably more well known than the n-factorial. Part of the reason for the popularity of prime factorization is its usefulness when calculating Greatest Common Factor (GCM) and the Least Common Multiplier (LCM), but we digress.
To understand what the factorial does or means, we should start with an example. We could choose any number n and calculate its n-factorial value, but it's best to choose a relatively small number, so let's use 5-factorial.
5! = 5 * 4 * 3 * 2 * 1 = 120
From this example, you can see that it is not rocket science, and you might even take a guess at the factorial formula. You can also understand why this exclamation point in maths can be regarded as an abbreviation since it is not a new operation but rather a collection of multiplications. In short, and somewhat informally, we can define the factorial as the multiplication of all the positive integers smaller than and equal to the given number.
Playing a bit with this, we can see that for 5-factorial, we can relate it to the 4-factorial in a straightforward way:
5! = 5 * (4 * 3 * 2 * 1) = 5 * 4!
This kind of relationship between n-factorials with a different n is the basis of the mathematical formula that defines the factorial operation, as we will see in the next section.
The factorial operation is not used everywhere in maths, but it is essential in statistics and probability problems. In those cases, especially when one has to deal with permutations or combinatorics, the n-factorial appears almost all the time. In the following sections of our factorial calculator, we will see real-world examples of problems that require the usage of factorials and the factorial formula.
So the factorial of n is denoted by n! and calculated by the product of integer numbers from 1 to n.
n! = 1×2×3×4×...×n
0! = 1
Factorial definition formula
1! = 1
2! = 1×2 = 2
3! = 1×2×3 = 6
4! = 1×2×3×4 = 24
5! = 1×2×3×4×5 = 120
Recursive factorial formula
n! = n×(n-1)!
5! = 5×(5-1)! = 5×4! = 5×24 = 120
5! ≈ √2π5⋅55⋅e-5 = 118.019
C program for factorial calculation
double factorial(unsigned int n)
if( n > 1 )
for(unsigned int k=2; k<=n; k++)
fact = fact*k;
Explanation of zero factorial
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