AHL 1.10 (HL)—Counting and extended binomial theorem

Count arrangements by deciding whether order matters.

Use the multiplication principle for sequential choices, permutations when order matters, and combinations when it does not.

Worked example
Choosing president and secretary from 5 uses 5P2=20; choosing a 2-person team uses 5C2=10.

Worked example
Does swapping two team members make a new team? no; use combinations.

Common boundary
Do not use permutations when the selected group has no roles.

Extended binomial form: for rational $n$, $(1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+\cdots$. When $n$ is negative or fractional this is an infinite expansion valid for $|x|<1$. For example, $(1+x)^{-1}=1-x+x^2-x^3+\cdots$. This objective excludes circular arrangements, identical-object permutations and proof of the theorem.