The Fundamental Counting
Principle
MATH 102
Contemporary Math
S. Rook
Overview
• Section 13.2 in the textbook:
– The fundamental counting principle
– Slot diagrams
The Fundamental Counting
Principle
The Fundamental Counting
Principle
• Recall that when situations can be broken up into
stages, we used a tree diagram to illustrate all
possibilities
– Can become cumbersome if we have many stages
or if one stage has many possibilities
• Instead, observe the pattern that results when
we use a tree diagram:
– e.g. A restaurant offers 2 appetizers, 3 entrées,
and 2 desserts. How many different meals can be
created if a meal consists of one appetizer, one
entrée, and one dessert?
The Fundamental Counting
Principle (Continued)
• This problem has how many stages/phases?
• How many choices do we have for each stage/phase?
• What is the relationship between the number of
choices for each stage/phase and the number of
possible meals?
• Fundamental Counting Principle: If a situation
can be broken up into stages where the first stage
has a choices, the second stage has b choices, the
third stage has c choices, etc, then the number of
total possibilities is a x b x c x …
– Do not forget to account for replacement if necessary
The Fundamental Counting
Principle (Example)
Ex 1: The board of an Internet start-up company
has seven members. If one person is to be in
charge of marketing and another in charge of
research, in how many ways can these two
positions be filled?
The Fundamental Counting
Principle (Example)
Ex 2: Elaine is building a home theater system
consisting of a tuner, an optical disc player,
speakers, and a high-definition TV. If she can
select from four tuners, eight speakers, three
optical disc players, and five TVs, in how many
ways can she configure her system?
The Fundamental Counting
Principle (Example)
Ex 3: Determine the total number of
possibilities when an 8-sided die and 12-sided
die are rolled simultaneously.
Slot Diagrams
Slot Diagrams
• A slot diagram is a visual way to illustrate the
Fundamental Counting Principle
• Useful in complex situations such as no
repetition
• Draw blanks for each stage of counting
– Write number of possibilities in the blank
– Multiply the possibilities together for all stages
• May have to examine sub problems before
applying the Fundamental Counting Principle
Slot Diagrams (Example)
Ex 4: In a certain state, license plates currently
consist of two letters followed by three digits.
a) How many such license plates are possible
if repeating is allowed?
b) How many such license plates are possible
if repeating is not allowed?
Slot Diagrams (Example)
Ex 5: To enter a vault, an authorized user must provide a 5
digit numeric PIN. How many different PINs exist if:
a) Digits are allowed to repeat?
b) Digits are not allowed to repeat?
c) The first digit is odd, the second digit is even, and the
rest of the digits are not allowed to repeat?
d) The PIN must be at least 59000 and digits are allowed
to repeat?
Slot Diagrams (Example)
Ex 6: Assume that we wish to seat three men:
Alex, Carl, and Frank and three women: Bonnie,
Daria, Edith in a row of six chairs. In how many
ways can we seat everyone if:
a) All the men sit together and all the women sit
together?
b) All the women sit together, but no woman sits
on an end seat?
c) Alex and Edith sit next to each other?
Slot Diagrams (Example)
Ex 7: Assume we wish to seat two men: Alex
and three women: Bonnie, Daria, and Edith in
a row of six chairs. In how many ways can we
seat everyone if the men sit together, the
women sit together AND there is exactly one
empty seat separating the two groups?
Summary
• After studying these slides, you should know
how to do the following:
– Use the Fundamental Counting Principle to solve a
variety of problems
• Additional Practice:
– See problems in Section 13.2
• Next Lesson:
– Permutations & Combinations (Section 13.3)