Section 3: Trees and Counting Techniques

Example Suppose a fast food restaurant sells ice cream cones in two sizes (regular and large) and three flavors (vanilla, chocolate, and strawberry). How many possible ice cream cones can be ordered?

Multiplication Rule of Counting  Fundamental Principle of Counting  If Task 1 can be done in n 1 ways and Task 2 can be done in n 2 ways, then Task 1 and Task 2 can be done in n 1 ∙n 2 ways.

Example A new car model is being produced by Limited Motors, Inc. It comes with a choice of two body styles, three interior package options, and four different colors, as well as the choice of automatic or standard transmissions. If a car dealership wants to carry one of each type of car, how many cars are required?

Ordered Arrangements Example How many different ways can eight cans of soup be displayed in a row?

! is read “factorial” 5! = “five factorial” = 5∙4∙3∙2∙1 = 120 Permutation the arrangement of objects in a certain order   (n – r)! P(n,r) = n! the number of permutations of n objects taken r at a time  P(n,n) = n! the number of permutations of n objects taken n at a time

Example Compute the number of ordered seating arrangements we have for eight people in five chairs. Example At the 1992 United States Olympic track and field trials only 4 of the 6 qualifiers were allowed to run in the race. How many different line ups were possible? Example Suppose a teacher wants to send 3 students to the board. In how many different orders can they go?

Combination: a collection of objects without regard to order and without repetition   r!(n – r)! C(n,r) = n! the number of combinations of n objects taken r at a time

Example Suppose a group of four students must pair up for a project. How many possible pairs are there?

Example Three members from a group of twelve on the board of directors at a Community Hospital will be selected to go to a convention with all expenses paid. How many different groups of three are there?