A tree diagram is a segmented graph in the shape of a tree in which no branch leads from any vertex back to itself. Each path through it represents a mutually exclusive event.

A tree diagram is a segmented graph in the shape of a tree in which no branch leads from any vertex back to itself. Each path through it represents a mutually exclusive event.

Tree DiagramTree Diagram

Calvary Christian Academy is having an election of student officers. Three students are running for president—Juan, Pam, and Jeff. There are two candidates for vice president—Doyle and Julianne. How many different ways are there to fill the offices?

Calvary Christian Academy is having an election of student officers. Three students are running for president—Juan, Pam, and Jeff. There are two candidates for vice president—Doyle and Julianne. How many different ways are there to fill the offices?

Example 1Example 1Find the number of ways that a student can select a two-digit number if the first digit must be odd and the second digit must be less than five.

Find the number of ways that a student can select a two-digit number if the first digit must be odd and the second digit must be less than five.

possible first digit—1, 3, 5, 7, 9possible first digit—1, 3, 5, 7, 9possible second digit—

0, 1, 2, 3, 4possible second digit—

0, 1, 2, 3, 4

0 1 2 3 40 1 2 3 4 0 1 2 3 40 1 2 3 4

0 1 2 3 40 1 2 3 40 1 2 3 40 1 2 3 40 1 2 3 40 1 2 3 4

11 33 55 77 99

30, 31, 32, 33, 3430, 31, 32, 33, 34

There are 25 different two-digit numbers.

There are 25 different two-digit numbers.

50, 51, 52, 53, 5450, 51, 52, 53, 5470, 71, 72, 73, 7470, 71, 72, 73, 7490, 91, 92, 93, 9490, 91, 92, 93, 94

10, 11, 12, 13, 1410, 11, 12, 13, 14

Make a tree diagram to find the number of combinations of three pairs of pants, three coats, and four shirts.

Make a tree diagram to find the number of combinations of three pairs of pants, three coats, and four shirts.

3636

ExampleExample

Make a tree diagram to find the number of possible milkshakes that could be ordered if chocolate and vanilla shakes are available in small, medium, and large.

Make a tree diagram to find the number of possible milkshakes that could be ordered if chocolate and vanilla shakes are available in small, medium, and large.

66

ExampleExample

Make a tree diagram to find the number of ways to make fifty cents in change using nickels, dimes, and quarters.

Make a tree diagram to find the number of ways to make fifty cents in change using nickels, dimes, and quarters.

1010

ExampleExample

Fundamental Principle of Counting

Fundamental Principle of Counting

If there are p ways that a first choice can be made and q ways that a second choice

can be made, then there are p × q ways to make the first

choice followed by the second choice.

If there are p ways that a first choice can be made and q ways that a second choice

can be made, then there are p × q ways to make the first

choice followed by the second choice.

Reid has five dress shirts and four ties. How many different shirt-and-tie combinations are possible?

Reid has five dress shirts and four ties. How many different shirt-and-tie combinations are possible?

5 × 4 5 × 4 = 20= 20

Example 2Example 2

How many different two-digit counting numbers can be formed if the first digit must be a nonzero even digit and the second digit must be less than seven but greater than zero?

How many different two-digit counting numbers can be formed if the first digit must be a nonzero even digit and the second digit must be less than seven but greater than zero?

Example 3Example 3

There are four choices (2, 4, 6, 8) for the first digit

and six choices (1, 2, 3, 4, 5, 6) for the

second digit.

There are four choices (2, 4, 6, 8) for the first digit

and six choices (1, 2, 3, 4, 5, 6) for the

second digit.

By the Fundamental Principle of Counting there are 4 x 6 =

24 such numbers.

By the Fundamental Principle of Counting there are 4 x 6 =

24 such numbers.

Mr. Dillard is buying a new car. He has the options given in the following table to choose from. How many different options does he have? If he chooses a white exterior, how many combinations does he have on the remaining options?

Mr. Dillard is buying a new car. He has the options given in the following table to choose from. How many different options does he have? If he chooses a white exterior, how many combinations does he have on the remaining options?

Example 4Example 4

Inte

rior

Inte

rior

redred

whitewhite

blackblack

silversilver

Tran

smis

sion

Tran

smis

sion

blackblack

blueblue

graygray

AM/FMAM/FM

+ CD+ CD

+ DVD+ DVD

automaticautomatic

manualmanualPac

kage

Packa

ge

Exte

rior

Exte

rior

How many different options does he have?

If he chooses a white exterior, how many combinations does he have on the remaining options?

How many different options does he have?

If he chooses a white exterior, how many combinations does he have on the remaining options?

4 × 3 × 3 × 24 × 3 × 3 × 2 = 72= 72

1 × 3 × 3 × 21 × 3 × 3 × 2 = 18= 18

Use the Fundamental Principle of Counting to find the number of possible three-digit area codes if the first number cannot be 0 or 1.

Use the Fundamental Principle of Counting to find the number of possible three-digit area codes if the first number cannot be 0 or 1.

800800

ExampleExample

How many different license plates are possible if three letters must be followed by three numbers?

How many different license plates are possible if three letters must be followed by three numbers?

17,576,00017,576,000

ExampleExample

How many different license plates are possible if none of the letters or numbers can repeat?

How many different license plates are possible if none of the letters or numbers can repeat?

11,232,00011,232,000

ExampleExample

How many ways can a family of four line up for a photograph?

How many ways can a family of four line up for a photograph?

2424

ExampleExample

How many combinations are possible on a school locker if the lock consists of the numbers 1 to 40 and the combination is a three-digit sequence of numbers?

How many combinations are possible on a school locker if the lock consists of the numbers 1 to 40 and the combination is a three-digit sequence of numbers?

64,00064,000

ExampleExample

How many combinations are possible if no two consecutive numbers are the same?

How many combinations are possible if no two consecutive numbers are the same?

60,84060,840

ExampleExample

How many ways can you seat five couples in a row of ten chairs, assuming, of course, that each couple is seated together?

How many ways can you seat five couples in a row of ten chairs, assuming, of course, that each couple is seated together?

3,8403,840

ExampleExample