6
SETS AND
COUNTING
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6.3
The Multiplication
Principle
Copyright © Cengage Learning. All rights reserved.
The Fundamental Principle of Counting
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The Fundamental Principle of Counting
The solution of certain problems requires more
sophisticated counting techniques.
We begin by stating a fundamental principle of counting
called the multiplication principle.
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Example 1
Three trunk roads connect Town A and Town B, and two
trunk roads connect Town B and Town C.
a. Use the multiplication principle to find the number of
ways in which a journey from Town A to Town C via
Town B can be completed.
b. Verify part (a) directly by exhibiting all possible routes.
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Example 1 – Solution
a. Since there are three ways of performing the first task
(going from Town A to Town B) followed by two ways of
performing the second task (going from Town B to Town
C), the multiplication principle says that there are 3  2,
or 6, ways to complete a journey from Town A to Town C
via Town B.
b. Label the trunk roads connecting Town A and Town B
with the Roman numerals I, II, and III, and label the trunk
roads connecting Town B and Town C with the
lowercase letters a and b.
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Example 1 – Solution
cont’d
A schematic of this is shown in Figure 12.
Roads from Town A to Town C
Figure 12
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Example 1 – Solution
cont’d
Then the routes from Town A to Town C via Town B can
be exhibited with the aid of a tree diagram (Figure 13).
Tree diagram displaying the possible
routes from Town A to Town C
Figure 13
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Example 1 – Solution
cont’d
If we follow all of the branches from the initial point A to the
right-hand edge of the tree, we obtain the six routes
represented by six ordered pairs:
(I, a), (I, b), (II, a), (II, b), (III, a), (III, b)
where (I, a) means that the journey from Town A to Town B
is made on Trunk Road I with the rest of the journey, from
Town B to Town C, completed on Trunk Road a, and so
forth.
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The Fundamental Principle of Counting
The multiplication principle can be easily extended, which
leads to the generalized multiplication principle.
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Example 3
A coin is tossed three times, and the sequence of heads
and tails is recorded.
a. Use the generalized multiplication principle to determine
the number of possible outcomes of this activity.
b. Exhibit all the sequences by means of a tree diagram.
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Example 3 – Solution
a. The coin may land in two ways. Therefore, in three
tosses, the number of outcomes (sequences) is given by
2  2  2, or 8.
b. Let H and T denote the outcomes “a head” and “a tail,”
respectively.
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Example 3 – Solution
cont’d
Then the required sequences may be obtained as shown
in Figure 14, giving the sequence as HHH, HHT, HTH,
HTT, THH, THT, TTH, and TTT.
Tree diagram displaying possible
outcomes of three consecutive coin tosses
Figure 14
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Practice
p. 343 Self-Check Exercises #1 & 2
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