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Study Guides > College Algebra

Summary: Counting Principles

Key Equations

number of permutations of nn distinct objects taken rr at a time P(n,r)=n!(n−r)!P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}
number of combinations of nn distinct objects taken rr at a time C(n,r)=n!r!(n−r)!C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}
number of permutations of nn non-distinct objects n!r1!r2!…rk!\frac{n!}{{r}_{1}!{r}_{2}!\dots {r}_{k}!}
Binomial Theorem (x+y)n=∑k−0n(nk)xn−kyk{\left(x+y\right)}^{n}=\sum _{k - 0}^{n}\left(\begin{array}{c}n\\ k\end{array}\right){x}^{n-k}{y}^{k}
(r+1)th\left(r+1\right)th term of a binomial expansion (nr)xn−ryr\left(\begin{array}{c}n\\ r\end{array}\right){x}^{n-r}{y}^{r}

Key Concepts

  • If one event can occur in mm ways and a second event with no common outcomes can occur in nn ways, then the first or second event can occur in m+nm+n ways.
  • If one event can occur in mm ways and a second event can occur in nn ways after the first event has occurred, then the two events can occur in m×nm\times n ways.
  • A permutation is an ordering of nn objects.
  • If we have a set of nn objects and we want to choose rr objects from the set in order, we write P(n,r)P\left(n,r\right).
  • Permutation problems can be solved using the Multiplication Principle or the formula for P(n,r)P\left(n,r\right).
  • A selection of objects where the order does not matter is a combination.
  • Given nn distinct objects, the number of ways to select rr objects from the set is C(n,r)\text{C}\left(n,r\right) and can be found using a formula.
  • A set containing nn distinct objects has 2n{2}^{n} subsets.
  • For counting problems involving non-distinct objects, we need to divide to avoid counting duplicate permutations.
  • (nr)\left(\begin{array}{c}n\\ r\end{array}\right) is called a binomial coefficient and is equal to C(n,r)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 mm ways and a second event with no common outcomes can occur in nn ways, then the first or second event can occur in m+nm+n ways binomial coefficient the number of ways to choose r objects from n objects where order does not matter; equivalent to C(n,r)C\left(n,r\right), denoted (nr)\left(\begin{array}{c}n\\ r\end{array}\right) binomial expansion the result of expanding (x+y)n{\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 mm ways and a second event can occur in nn ways after the first event has occurred, then the two events can occur in m×nm\times n ways; also known as the Multiplication Principle Multiplication Principle if one event can occur in mm ways and a second event can occur in nn ways after the first event has occurred, then the two events can occur in m×nm\times n ways; also known as the Fundamental Counting Principle permutation a selection of objects in which order matters

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