Permutations — The Fundamental Principle of Counting and Factorial Notation

Before permutations and combinations can be defined precisely, we need two basic tools.

The Fundamental Principle of Counting (multiplication principle). If one task can be performed in $m$ different ways, and after it is performed a second, independent task can be performed in $n$ different ways, then the two tasks together can be performed in $m \times n$ ways. This extends to any number of tasks performed in sequence: multiply the number of choices at each stage.

The addition principle. If a task can be performed in $m$ ways or, mutually exclusively, in $n$ ways (but not both), it can be performed in $m+n$ ways in total.

Example: a company has 3 shortlisted vendors for raw material and 4 shortlisted vendors for packaging. If one vendor of each type must be chosen, the number of ways to make both choices is $3 \times 4 = 12$ (multiplication principle, since both choices are made). If instead only one contract — either the material contract or the packaging contract — is being awarded to a single vendor, the number of ways is $3+4=7$ (addition principle, since the two vendor pools do not overlap).

Factorial notation. For a positive integer $n$, $n!$ (read "$n$ factorial") is the product of all positive integers from $1$ to $n$:

$n! = n(n-1)(n-2)\cdots 3 \cdot 2 \cdot 1$ …