## 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] |

## Key Concepts

- 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.

## Glossary

- 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

- combination
- a selection of objects in which order does not matter

- 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

- permutation
- a selection of objects in which order matters