Name ________________________
AP Statistics: Multiplication Rule AKA Fundamental Counting Theorem
Fundamental Counting Principle (Multiplication Principle):
If there are m ways to do one thing and n ways to do another, then there are
m•n ways of doing both in sequence.
This rule can be extended to any number of events happening in sequence.
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Example 1: It’s time for the rock ’n’ roll quartet – the Fab Four – to have their photo taken
for their latest CD cover. The leader of the band tells the photographer, “Hey, pal, let’s get
started. I want to see every possible arrangement of the group before I decide on the final
cover shot.” How many different ways the band members can be arranged if they are
standing shoulder to shoulder?
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Example 2: Mrs. Blubaugh has decided that for her daughters’ birthday she will buy her a
“grown-up” doll. She has found that a popular doll company offers the ability to design your
own doll. You can choose from 3 hair colors, 3 eyes colors, and 2 dress colors. How many
possible dolls are there?
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Example 3: A die is tossed and a card is drawn from a 52-card deck. How many outcomes
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Example 4: Student ID’s are 5 characters long, consisting of 5 digits. How many student
are possible?
ID’s are possible?
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Example 5: In a local restaurant, there are 3 choices for appetizer, 7 choices for entrée, 4
items for dessert, and 5 choices for drinks. In how many ways can a customer order a full
meal?
1. For each of the following, use a tree diagram or the multiplication principle to determine
the number of outcomes in the sample space. Then write the sample space using set
notation.
a) Toss 2 coins.
b) Toss 3 coins.
c) Toss 4 coins.
For each of the following, use multiplication principle to determine the number of outcomes
in the sample space.
2. Suppose a county license tag has a four-digit number for identification.
a) If any digit can occupy any of the four positions, how many county license tags can you
have?
b) If the county license tags described in a) do not allow duplicate digits, how many county
license tags can you have?
3. Suppose a state’s license plates have three digits, then the state seal, and then three
more digits.
a) If any digit can go in any position, how many license plates are possible?
b) Suppose the state used up all of the six-digit combinations and decided to replace the
first three digits with letters of the alphabet. How many license plates are now possible?
c) The state had to expand once more and added a space for a fourth digit at the end.
Example: ABC 1234. How many license plates are possible?
d) Another state’s license plate contains two letters of the alphabet and four digits.
Calculate the number of license plates possible for this state.
e) Why do you think states use different formats for creating their license plates?
4. Telephone Numbers: In the last twenty years, Loudoun County has had tremendous
growth, not only in population, but also in the number of cell phones and internet accessible
computers. This growth has resulted in a shortage of telephone numbers for the local area
code of 703.
a) How many phone numbers are possible with the (703) area code?
b) The telephone commission has hired you as a consultant to advise them on the
number of telephone numbers that would available if they added another area code,
(571), to the region. Your job is to calculate that number. Your calculation is based
on seven digits for the main number. How many new numbers are there? How many
total numbers does Loudoun County have now?
c) After reviewing your estimate, the telephone commission returns your report and
asks you to make adjustments. They ask you to recalculate the number of available
telephone numbers, given that the first two digits of the main seven digit number
must not contain a zero or a one.
d) How many possible area codes can be created if the first digit cannot be a zero or
one?