# Tree Diagrams and Counting Principles Worksheet

Tree Diagrams and Counting Principles Worksheet
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1.
Luke has 3 CDs and 3 DVDs with 3 songs recorded in each of them. In how many ways can he play a song?
 a. 27 b. 24 c. 36 d. 54

#### Solution:

Number of CDs with Luke = 3

Number of DVDs with Luke = 3

Number of ways of playing a song = (Number of CDs with Luke) × (Number of DVDs with Luke) × (Number of songs in each CD or DVD)

= 3 × 3 × 3 = 27
[Substitute and multiply.]

There are 27 different ways in which Luke can play a song.

2.
There are 3 Spanish, 3 German and 3 French books in a shelf. In how many ways can one Spanish, one German and one French book be chosen?
 a. 36 b. 48 c. 54 d. 27

#### Solution:

Number of ways of choosing a Spanish book = 3

Number of ways of choosing a German book = 3

Number of ways of choosing a French book = 3

Number of ways of choosing one Spanish, one German and one French book = (Number of ways of choosing a Spanish book) × (Number of ways of choosing a German book) × (Number of ways of choosing a French book)

= 3 × 3 × 3 = 27
[Substitute and multiply.]

There are 27 ways to choose a Spanish book, a German book and a French book.

3.
A bag contains 3 yellow balls and 4 pink balls. Kay picked two balls one after the other. Which tree diagram shows all the possible outcomes?

 a. Tree diagram 4 b. Tree diagram 2 c. Tree diagram 3 d. Tree diagram 1

4.
Lara has 3 soft toys, 3 cards and 3 gift items. Find the number of ways of choosing a soft toy, a card and a gift item.
 a. 3 b. 9 c. 6 d. 27

#### Solution:

Number of ways of choosing a soft toy = 3

Number of ways of choosing a card = 3

Number of ways of choosing a gift item = 3

Number of ways of choosing a soft toy, a card and a gift item = (Number of ways of choosing a soft toy) × (Number of ways of choosing a card) × (Number of ways of choosing a gift item)

= 3 × 3 × 3 = 27
[Substitute and multiply.]

There are 27 ways to choose a soft toy, a card and a gift item.

5.
A balloon-seller has 3 different shapes of balloons in 3 different sizes (small, medium, and large) and 3 different colors (orange, red, and blue). In how many possible ways can you choose a balloon?
 a. 12 b. 27 c. 36 d. 54

#### Solution:

Number of choices of shapes = 3

Number of choices of sizes = 3

Number of choices of colors = 3

Number of ways of choosing a balloon = (Number of choices of shapes) × (Number of choices of sizes) × (Number of choices of colors

= 3 × 3 × 3 = 27
[Substitute and multiply.]

There are 27 different ways in which you can choose a balloon.

6.
A restaurant offers three choices of appetizers (Spinach bread, Garlic bread, and Cheese bread), three choices of juices (Orange, Lemon, and Apple) and three choices of salads (Onion salad, Spinach salad, and Carrot salad). If you want to have an appetizer, a juice, and a salad, in how many ways can you choose them?
 a. 9 b. 3 c. 12 d. 27

#### Solution:

Number of choices of appetizers = 3

Number of choices of juices = 3

Number of choices of salads = 3

Number of ways of choosing an appetizer, a juice and a salad = (Number of choices of appetizers) × (Number of choices of juices) × (Number of choices of salads)

= 3 × 3 × 3 = 27
[Substitute and multiply.]

There are 27 different ways in which you can choose an appetizer, a juice, and a salad.

7.
If you choose a medal at random, then what is the probability that you choose a gold medal?

 a. $\frac{1}{7}$ b. $\frac{1}{4}$ c. $\frac{1}{8}$ d. $\frac{1}{2}$

#### Solution:

Number of different ways of choosing a medal = (Number of ways of choosing a category) × (Number of ways of choosing a team) × (Number of ways of choosing a medal)

= 2 × 2 × 2 = 8
[Substitute and multiply.]

P(gold medal) = Number of favorable outcomes / Total number of possible outcomes

Number of favorable outcomes = Number of gold medals given = 4

Total number of possible outcomes = Number of ways of choosing a medal = 8

= 48 = 12
[Substitute and simplify.]

Probability of choosing a gold medal is 1 / 2.

8.
A blue, red or green cube is selected and a coin is tossed. Which of the tree diagrams represents all the possible outcomes of the two events?

 a. Tree diagram 1 b. Tree diagram 2 c. Tree diagram 3 d. Tree diagram 4

#### Solution:

The outcome of first event is a red, blue or a green cube.

The outcome of second event is a head or a tail.

The outcome of the two events is an outcome of head or a tail for every outcome of a red, blue or a green cube.

The possible combinations of the two events are (Blue, Head), (Blue, Tail), (Red, Head), (Red, Tail), (Green, Head) and (Green, Tail) as represented in the tree diagram 1.

The tree diagram 1 has all the possible combinations of the two events.

9.
There are 20 English books and 7 French books in a shelf. In how many ways can one English book and one French book be chosen?
 a. 140 b. 7 c. 20 d. 2

#### Solution:

Number of ways of choosing an English book = 20

Number of ways of choosing a French book = 7

Number of ways of one English and one French book can be chosen = (Number of ways of choosing an English book) × (Number of ways of choosing a French book)

= 20 × 7 = 140
[Substitute and multiply.]

There are 140 ways to choose an English book and a French book.

10.
Using a tree diagram, find the number of ways in which 3 prizes (first, second and third ) can be distributed to the 4 students, John, Mike, Joseph, and Kevin who participated in an essay writing competition.
 a. 4 b. 9 c. 12 d. 6

#### Solution:

Each student can get 1st, 2nd or 3rd prize.

Make a tree diagram, which represents the prizes given to each of the students as shown below.

The branches in the tree diagram represent the different ways in which the prizes can be distributed.

So, there are 12 possible ways in which the prizes can be distributed to the students.