Key Equations

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

Key Concepts

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

Glossary

if one event can occur in $m$ ways and a second event with no common outcomes can occur in $n$ ways, then the first or second event can occur in $m+n$ ways
if one event can occur in $m$ ways and a second event can occur in $n$ ways after the first event has occurred, then the two events can occur in $m\times n$ ways; also known as the Multiplication Principle
if one event can occur in $m$ ways and a second event can occur in $n$ ways after the first event has occurred, then the two events can occur in $m\times n$ ways; also known as the Fundamental Counting Principle