(problem 1a) Sam is getting dressed and wants to include a belt and a hat. If Sam
has three distinct belts and two distinct hats, how many ways can Sam accessorize?
Include a tree diagram that illustrates the possibilities.
Sam can accessorize in different ways.
We use the Fundamental Principle of Counting.
1 Fundamental Principle of Counting
example 1 Pat has three different shirts and two different pairs of pants. Getting
dressed requires one of each. In how many ways can Pat get dressed?
We make a tree diagram to determine the possible outfits Pat can create. 
We make a tree diagram to determine the possible outfits Pat can create.
Each path from Pat to a node on the far right of the tree diagram represents a different way for Pat to dress. From top to bottom, these represent the following outfits: Shirt 1 with Pants 1, Shirt 1 with Pants 2, Shirt 2 with Pants 1, Shirt 2 with Pants 2, Shirt 3 with Pants 1 and Shirt 3 with Pants 2. Since there are 6 total paths, Pat can dress in 6 different ways.
(problem 1b) A couple wants to go on a date. They decide to eat at a restaurant,
walk in a park and then see a show. If there are two restaurants that they both like,
two nearby parks and two shows playing, how many possibilities are there for their
date? Make a tree diagram illustrating the possibilities.
There are different possibilities for their date.
There are different possibilities for their date.
Fundamental Principle of Counting Suppose that decisions are to be made and that
the number of choices for the decision is . Then the total number of ways to make
the decisions is
example 2 There are three types of Tesla Model 3: Standard Range, Long Range and
Performance. Each type can be painted one of 5 colors. Furthermore, two interior
color schemes are available. Given these parameters, how many choices are there for
my Tesla Model 3 purchase?
In total, three decisions must be made: the type, the paint color and the interior color scheme. The number of choices for each decision is 3, 5 and 2 respectively. Hence, by the Fundamental Principle of Counting, the total number of ways to configure my new Tesla Model 3 is . 
In total, three decisions must be made: the type, the paint color and the interior color scheme. The number of choices for each decision is 3, 5 and 2 respectively. Hence, by the Fundamental Principle of Counting, the total number of ways to configure my new Tesla Model 3 is .
(problem 2a) A three course meal consists of an appetizer, an entree and a dessert. If
a restaurant offers a dozen appetizers, a score of entrees and a bakers dozen desserts,
how many possible three course meals can be ordered?
The total number of possible three course meals is .
The total number of possible three course meals is .
(problem 2b) A sixth grade class of 20 students would like to select a President,
Vice-President, Secretary and Treasurer. If no student may serve in more than one
post, in how many possible ways can the class officers be selected?
The total number of ways to select the class officers is .
The total number of ways to select the class officers is .
example 3 Suppose license plates consist of three letters followed by three digits. How
many license plates are possible?
Altogether, there are six decisions to be made. By the Fundamental Principle of Counting, the answer is the product of the number of choices for each decision. 
Thus the total number of such license plates is .
Altogether, there are six decisions to be made. By the Fundamental Principle of Counting, the answer is the product of the number of choices for each decision.
(problem 3) Suppose telephone numbers consist of 7 digits, the first of which cannot
be 0 or 1. How many such telephone numbers are possible?
example 4 Computer passwords are eight characters long where a character can be
either an upper case letter, lower case letter, digit 0-9, or one of 8 special symbols.
How many passwords are possible?
There are 26+26+10+8 = 70 possibilities for each character in the password. Since there are 8 characters in the password, the FPC says that there are (8 times) passwords. Hence the total number of possible passwords is . Typically there are restrictions on passwords that will limit the total number possible, and we will continue to explore these scenarios as we move forward.
There are 26+26+10+8 = 70 possibilities for each character in the password. Since there are 8 characters in the password, the FPC says that there are (8 times) passwords. Hence the total number of possible passwords is . Typically there are restrictions on passwords that will limit the total number possible, and we will continue to explore these scenarios as we move forward.
(problem 4) Computer passwords consist of 6 characters, where a character can be
either a lower case letter, a digit 1-9, or one of 4 special symbols. How many such
passwords are possible?
The number of possible passwords is .
The number of possible passwords is .
example 5 A pair of 6-sided dice, one red and the other green, are thrown. The
numbers on the top face of each die are recorded in the form (red, green). How many
outcomes are possible?
Since there are 6 possible outcomes for each die roll (1-6), the FPC says that the total number of possible outcomes is .
Since there are 6 possible outcomes for each die roll (1-6), the FPC says that the total number of possible outcomes is .
(problem 5) Three 4-sided dice (numbered 1-4), colored red, green and blue, are
thrown. The numbers on each die are recorded in the format (red, green, blue). How
many outcomes are possible?
The total number of possible outcomes for the three dice in the indicated format is .
The total number of possible outcomes for the three dice in the indicated format is .
example 6 A bit matrix is a matrix (rectangular array of numbers) whose entries are
zeros and ones. How many bit matrices are there?
A matrix has 4 entries, and with 2 choices for each entry, the FPC says that the total number of such matrices is .
2024-09-27 14:05:35 A matrix has 4 entries, and with 2 choices for each entry, the FPC says that the total number of such matrices is .