Summary: Counting Principles

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]
Binomial Theorem [latex]{\left(x+y\right)}^{n}=\sum _{k - 0}^{n}\left(\begin{array}{c}n\\ k\end{array}\right){x}^{n-k}{y}^{k}[/latex]
[latex]\left(r+1\right)th[/latex] term of a binomial expansion [latex]\left(\begin{array}{c}n\\ r\end{array}\right){x}^{n-r}{y}^{r}[/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.
  • [latex]\left(\begin{array}{c}n\\ r\end{array}\right)[/latex] is called a binomial coefficient and is equal to [latex]C\left(n,r\right)[/latex].
  • 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 [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

binomial coefficient the number of ways to choose r objects from n objects where order does not matter; equivalent to [latex]C\left(n,r\right)[/latex], denoted [latex]\left(\begin{array}{c}n\\ r\end{array}\right)[/latex]

binomial expansion the result of expanding [latex]{\left(x+y\right)}^{n}[/latex] 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 [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