• Introduction
• Iteration
  • Summation
  • Product
  • Set Notation
  • Nested Iteration
  • Conditions on indices
  • Union and Intersection
  • General Iteration
• General Symbols
  • Cases
  • Function Composition
  • Set Containment
  • Kronecker
• Operations
  • Factorial
  • Binomial
• Differentiation
  • Derivative
  • Partial Derivative
  • Gradient

Factorial: $n!$

The factorial of a number is specified by an exclamation mark. It is the number of different ways you can order n elements.

It is the product of the first n positive natural numbers, so it is an integer.

$$n! = 1 \cdot 2 \cdot 3 \cdot \dots \cdot n$$

Using the product symbol as seen in another section we can also write it as:

$$n! = \prod\limits_{i=1}^n i$$

From this we also get $0! = 1$. It is not defined for negative values.

Attention: The factorial increases very rapidly. If you use a 32 bit signed integer, then $13!$ will already overflow. Even for a 64 bit signed integer, $21!$ is too much. So if you see a formula involving factorials, try to cancel out terms as much as possible. For example, if $k<n$: $\frac{n!}{k!} = (k+1)(k+2) \cdots (n-1) n$

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