Section 4.2 Counting
253
Example 24 illustrates that the total number of outcomes for a sequence
of events can be obtained by multiplying the number of outcomes for the first
event by the number of outcomes for the second. This idea is summarized in the
multiplication principle.
pRinciple mUltiPliCatiON PRiNCiPle
If there are n 1 possible outcomes for a first event and n 2 possible outcomes for
a second event, there are n 1 × n 2 possible outcomes for the sequence of the two
events.
The multiplication principle can be extended by induction to apply to a
sequence of any finite number of events. (See Exercise 73 at the end of this section.) The multiplication principle is useful whenever we want to count the total
number of possible outcomes for a task that can be broken down into a sequence
of successive subtasks.
ReminDeR
Use the multiplication
principle when there is a
sequence of events.
example 25
The last part of your telephone number contains four digits. How many such
four-digit numbers are there?
We can construct four-digit numbers by performing a sequence of subtasks:
choose the first digit, then the second, the third, and finally the fourth. The first
digit can be any one of the 10 digits from 0 to 9, so there are 10 possible outcomes
for the first subtask. Likewise, there are 10 different possibilities each for the second digit, the third, and the fourth. Using the multiplication principle, we multiply
the number of outcomes for each subtask in the sequence. Therefore there are
10 # 10 # 10 # 10 = 10,000 different numbers.
If an element cannot be used again—that is, if repetitions are not allowed—
the number of possible outcomes for successive events will be affected.
example 26
Referring to Example 25, how many four-digit numbers are there if the same digit
cannot be used twice?
Again we have the sequence of subtasks of selecting the four digits, but no
repetitions are allowed. There are 10 choices for the first digit, but only 9 choices
for the second because we can’t use what we used for the first digit, and so on.
There are 10 # 9 # 8 # 7 = 5040 different numbers.
example 27
a. How many ways are there to choose three officers from a club of
25 people?
b. How many ways are there to choose three officers from a club of 25 people
if someone can hold more than one office?
In (a), there are three successive subtasks with no repetitions. The first subtask,
choosing the first officer, has 25 possible outcomes. The second subtask has 24 outcomes, the third 23 outcomes. The total number of outcomes is 25 # 24 # 23 = 13,800.
In (b), the same three subtasks are done in succession, but repetitions are allowed.
The total number of outcomes is 25 # 25 # 25 = 15,625.
253
Example 24 illustrates that the total number of outcomes for a sequence
of events can be obtained by multiplying the number of outcomes for the first
event by the number of outcomes for the second. This idea is summarized in the
multiplication principle.
pRinciple mUltiPliCatiON PRiNCiPle
If there are n 1 possible outcomes for a first event and n 2 possible outcomes for
a second event, there are n 1 × n 2 possible outcomes for the sequence of the two
events.
The multiplication principle can be extended by induction to apply to a
sequence of any finite number of events. (See Exercise 73 at the end of this section.) The multiplication principle is useful whenever we want to count the total
number of possible outcomes for a task that can be broken down into a sequence
of successive subtasks.
ReminDeR
Use the multiplication
principle when there is a
sequence of events.
example 25
The last part of your telephone number contains four digits. How many such
four-digit numbers are there?
We can construct four-digit numbers by performing a sequence of subtasks:
choose the first digit, then the second, the third, and finally the fourth. The first
digit can be any one of the 10 digits from 0 to 9, so there are 10 possible outcomes
for the first subtask. Likewise, there are 10 different possibilities each for the second digit, the third, and the fourth. Using the multiplication principle, we multiply
the number of outcomes for each subtask in the sequence. Therefore there are
10 # 10 # 10 # 10 = 10,000 different numbers.
If an element cannot be used again—that is, if repetitions are not allowed—
the number of possible outcomes for successive events will be affected.
example 26
Referring to Example 25, how many four-digit numbers are there if the same digit
cannot be used twice?
Again we have the sequence of subtasks of selecting the four digits, but no
repetitions are allowed. There are 10 choices for the first digit, but only 9 choices
for the second because we can’t use what we used for the first digit, and so on.
There are 10 # 9 # 8 # 7 = 5040 different numbers.
example 27
a. How many ways are there to choose three officers from a club of
25 people?
b. How many ways are there to choose three officers from a club of 25 people
if someone can hold more than one office?
In (a), there are three successive subtasks with no repetitions. The first subtask,
choosing the first officer, has 25 possible outcomes. The second subtask has 24 outcomes, the third 23 outcomes. The total number of outcomes is 25 # 24 # 23 = 13,800.
In (b), the same three subtasks are done in succession, but repetitions are allowed.
The total number of outcomes is 25 # 25 # 25 = 15,625.