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Study Guides > College Algebra

Key Concepts & Glossary

Key Equations

number of permutations of nn distinct objects taken rr at a time P(n,r)=n!(nr)!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!(nr)!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}!}

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.

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
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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