Summary: Counting Principles

Key Equations

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

Binomial Theorem	${\left(x+y\right)}^{n}=\sum\limits _{k - 0}^{n}\left(\begin{gathered}n\\ k\end{gathered}\right){x}^{n-k}{y}^{k}$
$\left(r+1\right)th$ term of a binomial expansion	$\left(\begin{gathered}n\\ r\end{gathered}\right){x}^{n-r}{y}^{r}$

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.
$\left(\begin{gathered}n\\ r\end{gathered}\right)$ is called a binomial coefficient and is equal to $C\left(n,r\right)$ .
The Binomial Theorem allows us to expand binomials without multiplying.
We can find a given term of a binomial expansion without fully expanding the binomial.

Glossary

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

binomial coefficient the number of ways to choose r objects from n objects where order does not matter; equivalent to $C\left(n,r\right)$ , denoted $\left(\begin{gathered}n\\ r\end{gathered}\right)$

binomial expansion the result of expanding ${\left(x+y\right)}^{n}$ by multiplying

Binomial Theorem a formula that can be used to expand any binomial

combination a selection of objects in which order does not matter

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

permutation a selection of objects in which order matters

Course Enrichment Unit