Key Concepts & Glossary

Key Equations

number of permutations of [latex]n[/latex] distinct objects taken [latex]r[/latex] at a time	[latex]P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}[/latex]
number of combinations of [latex]n[/latex] distinct objects taken [latex]r[/latex] at a time	[latex]C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}[/latex]
number of permutations of [latex]n[/latex] non-distinct objects	[latex]\frac{n!}{{r}_{1}!{r}_{2}!\dots {r}_{k}!}[/latex]

If one event can occur in [latex]m[/latex] ways and a second event with no common outcomes can occur in [latex]n[/latex] ways, then the first or second event can occur in [latex]m+n[/latex] ways.
If one event can occur in [latex]m[/latex] ways and a second event can occur in [latex]n[/latex] ways after the first event has occurred, then the two events can occur in [latex]m\times n[/latex] ways.
A permutation is an ordering of [latex]n[/latex] objects.
If we have a set of [latex]n[/latex] objects and we want to choose [latex]r[/latex] objects from the set in order, we write [latex]P\left(n,r\right)[/latex].
Permutation problems can be solved using the Multiplication Principle or the formula for [latex]P\left(n,r\right)[/latex].
A selection of objects where the order does not matter is a combination.
Given [latex]n[/latex] distinct objects, the number of ways to select [latex]r[/latex] objects from the set is [latex]\text{C}\left(n,r\right)[/latex] and can be found using a formula.
A set containing [latex]n[/latex] distinct objects has [latex]{2}^{n}[/latex] subsets.
For counting problems involving non-distinct objects, we need to divide to avoid counting duplicate permutations.

Addition Principle: if one event can occur in [latex]m[/latex] ways and a second event with no common outcomes can occur in [latex]n[/latex] ways, then the first or second event can occur in [latex]m+n[/latex] ways

Fundamental Counting Principle: if one event can occur in [latex]m[/latex] ways and a second event can occur in [latex]n[/latex] ways after the first event has occurred, then the two events can occur in [latex]m\times n[/latex] ways; also known as the Multiplication Principle

Multiplication Principle: if one event can occur in [latex]m[/latex] ways and a second event can occur in [latex]n[/latex] ways after the first event has occurred, then the two events can occur in [latex]m\times n[/latex] ways; also known as the Fundamental Counting Principle