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Chapter 13 sec 2
THE FUNDAMENTAL
COUNTING PRINCIPLE
Form your groups,
Please.
What is the FCP?
 Definition:
 If we want to perform a series of tasks and
the first task can be done in a ways, the
second can be done in b ways, the third can
be done in c ways, and so on, then all the
tasks can be done in
a x b x c x d x ……. ways.
Coin and dice problems.
 How many ways can four coins be
flipped?
 Solution:
 There are 4 tasks, flipping the
first coin in 2 ways. Therefore the
second, third, & fourth also can be
flipped 2 ways. Using FCP…
 2 x 2 x 2 x 2 = 16 ways
How many ways can four dice be
rolled?
 Solution:
 The first die can be rolled 6
ways. Using the FCP…
 6 x 6 x 6 x 6 = 1296 ways.
Methods of FCP
 1. Using the slot diagrams.
 2. Using the tree diagrams.
This chapter focuses on
the slot diagram.
Slot Diagram
 The slot diagram is a useful
technique for solving
problems to keep track of
the number of ways to do
each tasks.
Slot diagram
1st task
# of
ways
2nd
x
# of
ways
3rd
x
# of
ways
4th
x
# of
ways
5th
x
# of
ways
Breaking and Entering
(forgetfulness)
 Your office has a 5 digit
keypad lock. The numbers are
from 0, 1, …, 9. How many
different keypads patterns
are possible if …
 a) any digits can be used in any
position and repetition of digits
is allowed?
 b) the digit 0 cannot be using as
the first digit, but otherwise any
digit can be used in any position
and repetition is allowed.
 c) any digits can be used in any
position, but repetition is not
allowed.
Solution
A) 5 tasks and there are 10 digits.
1st task
10
2nd
x
10
3rd
x
= 100, 000 ways.
10
4th
x
10
5th
x
10
 b) 9 x 10 x 10 x 10 x 10
= 90,000 ways.
 c) 10 x 9 x 8 x 7 x 6
= 30,240 ways.
Group problems
 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?
 Solution: 42
 2. If the Chamber of
Commerce has 20 members, in
how many ways can they elect
a different president, vice
president, and treasurer?
 Solution: 6,840
 Using the digits 0, 1, 2, …, 9, determine
the four digit number of possibilities
for each situation.
 1. The number must begin and end with
an odd digit; digits may not be
repeated.
 Solution: 1,120
 The number must be
between 5,001 and 8,000;
digits may not be repeated.
 Solution: 1,512
 In a certain state, license plates
currently consist of two letters
followed by three digits. How many
such license plates are possible? If
the state department of
transportation decides to change the
plates to have three letters followed
by two digits, how many plates will
be possible?
 Solution: 676,000; 1,757,600