Thinking
Mathematically
The Fundamental Counting Principle
The Fundamental Counting Principle
If you can choose one item from a group of
M items and a second item from a group of
N items, then the total number of two-item
choices is M  N.
Example Applying the Fundamental
Counting Principle
The Greasy Spoon Restaurant offers 6
appetizers and 14 main courses.
Using the fundamental counting principle,
there are 14  6 = 84 different ways a
person can order a two-course meal.
Example Applying the Fundamental
Counting Principle
This is the semester that you decide to take your
required psychology and social science courses.
Because you decide to register early, there are 15
sections of psychology from which you can
choose. Furthermore, there are 9 sections of social
science that are available at times that do not
conflict with those for psychology. In how many
ways can you create two-course schedules that
satisfy the psychology-social science requirement?
Solution
The number of ways that you can satisfy the
requirement is found by multiplying the
number of choices for each course. You can
choose your psychology course from 15
sections and your social science course
from 9 sections. For both courses you have:
15  9, or 135 choices.
The Fundamental Counting
Principle
The number of ways a series of successive
things can occur is found by multiplying the
number of ways in which each thing can
occur.
Example Options in Planning a Course
Schedule
Next semester you are planning to take three
courses - math, English, and humanities. Based
on time blocks and highly recommended
professors, there are 8 sections of math, 5 of
English, and 4 of humanities that you find
suitable. Assuming no scheduling conflicts, there
are:
8  5  4 = 160 different three course schedules.
Example
Car manufacturers are now experimenting with
lightweight three-wheeled cars, designed for a
driver and one passenger, and considered ideal for
city driving. Suppose you could order such a car
with a choice of 9 possible colors, with or without
air-conditioning, with or without a removable
roof, and with or without an onboard computer. In
how many ways can this car be ordered in terms of
options?
Solution
This situation involves making choices with
four groups of items.
color - air-conditioning - removable roof - computer
9  2  2  2 = 72
Thus the car can be ordered in 72 different
ways.
Example A Multiple Choice Test
You are taking a multiple-choice test that
has ten questions. Each of the questions has
four choices, with one correct choice per
question. If you select one of these options
per question and leave nothing blank, in
how many ways can you answer the
questions?
Solution
We use the Fundamental Counting Principle
to determine the number of ways you can
answer the test. Multiply the number of
choices, 4, for each of the ten questions
4444444444
=1,048,576
Example Telephone Numbers in the
United States
Telephone numbers in the United States
begin with three-digit area codes followed
by seven-digit local telephone numbers.
Area codes and local telephone numbers
cannot begin with 0 or 1. How many
different telephone numbers are possible?
Solution
We use the Fundamental Counting Principle
to determine the number of different
telephone numbers that are possible.
8  10  10  8  10  10  10  10  10  10
=6,400,000,000
Thinking
Mathematically
The Fundamental Counting Principle