Key Concepts & Glossary

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	$\frac{n!}{{r}_{1}!{r}_{2}!\dots {r}_{k}!}$

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.

Addition Principle: 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

Fundamental Counting 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 Multiplication Principle

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