Lesson 1

Lesson 1: Fundamental Counting Principle

 

Focus

 

Prior to 1999, the area code for the province of Alberta was 403. In 1999, another area code (780) was added and became the area code for the northern part of the province. The southern half of the province retained the 403 area code. In September 2008, Alberta added another area code, 587, across the province. Adding another area code provides the potential for 9 000 000 more phone numbers. Can you think of why the province continues to need more phone numbers?

 

By the end of this lesson, you will be able to show that this change does indeed amount to 9 000 000 new phone numbers.

 

Graham/25113263/flickr

 

Historically, licence plates in Alberta have consisted of three letters followed by three digits. Because the province is getting short of licence-plate numbers, Alberta is now changing the format to three letters followed by four digits. In 1997, Ontario switched from three digits and three letters to four letters and three digits. Can you figure out how many new licence plates each of these scenarios will create?

 

Determining the number of ways in which something may occur is a branch of mathematics called combinatorics. Combinatorics is used in many ways: from designing a computer network to coding a garage door opener to determining the chances of winning the lottery.

 

Lesson Outcomes

 

At the end of this lesson you will be able to

• use the fundamental counting principle to solve problems
• use a new notation, called factorial notation, to determine numbers of arrangements of items

Lesson Questions

 

You will investigate the following questions:

• How can you use the fundamental counting principle to determine the number of outcomes in a given problem?
• What is factorial notation, and when is it used to solve problems?

Assessment

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 1 Assignment (Download the Lesson 1 Assignment and save it in your course folder now.)
• course folder submissions from Try This and Share activities
• additions to Glossary Terms and Formula Sheet
• work under Project Connection

Self-Check activities are for your own use. You can compare your answers to suggested answers to see if you are on track. If you are having difficulty with concepts or calculations, contact your teacher.

 

Remember that the questions and activities you will encounter provide you with the practice and feedback you need to successfully complete this course. You should complete all questions and place your responses in your course folder. Your teacher may wish to view your work to check on your progress and to see if you need help.

 

Time

 

Each lesson in Mathematics 30-2 Learn EveryWare is designed to be completed in approximately two hours. You may find that you require more or less time to complete individual lessons. It is important that you progress at your own pace, based on your individual learning requirements.

 

This time estimation does not include time required to complete Going Beyond activities or the Module Project.

Discover

 

When you solved probability problems in Module 2, you had to know the total number of possible outcomes in order to determine the probability of an event. In some cases, the total number of outcomes was obvious, as in the number of cards in a deck or the number of faces on a cube. In other cases, determining the number of outcomes was more difficult.

 

Try This 1

 

Louie’s Bistro offers a special three-course meal for $20.00. You have a choice of 2 salads—mixed or Caesar—and a choice of 4 entrées: chili, club sandwich, roasted chicken, or grilled fish. Finally, for dessert, you may choose a brownie, a cupcake, or a piece of cheesecake.

 

How many different three-course meals are possible?

 

Save your responses in your course folder.

 

Share 1

 

With a partner or in a group, discuss whether there is a more efficient way for determining the number of meals rather than writing out all possible meal combinations.

 

If required, save a copy of your discussion in your course folder.

Explore

 

In Share 1, someone may have suggested using a diagram to solve the problem. The following diagram is called a tree diagram.

 

 

Adapted from © gollli/26357551/Fotolia

 

By counting the total number of terminal branches in the tree diagram, you can see that you have a choice of 24 three-course meals. The list of all 24 possible meals is called the sample space for this problem. Remember from Module 2 that the sample space is a list of all possible outcomes. You could also list all possible outcomes in an outcome table. A partial table is shown.

 

Salad	Entrée	Dessert
mixed	chili	brownie
mixed	chili	cupcake
mixed	chili	cheesecake
mixed	club sandwich	brownie
mixed	club sandwich	cupcake
mixed	club sandwich	cheesecake

 

The first example on Tree Diagram at Mathematics Glossary demonstrates how all possible outcomes of tossing a penny and rolling a die can be calculated using a tree diagram. Note that the order of tossing the penny or rolling the die does not affect the outcomes or sample space.

 

 

In the next Try This, you will find the possible outcomes using tree diagrams and outcome tables. You will then determine an expression to calculate these outcomes.

 

Try This 2

In “Investigate the Math” question G, which you completed as part of Try This 2, you were asked to create an expression relating the number of routes between

• Winnipeg and Regina, a
• Regina and Saskatoon, b

For each of the 3 routes between Winnipeg and Regina, there were 2 routes from Regina to Saskatoon. This is represented by a • b. That means that there were 3 × 2 or 6 ways from Winnipeg to Saskatoon, initially, and 4 × 3 or 12 ways once the new routes were added.

 

 

 

 

 

 

 

 

Think back to the example from Discover at the start of this lesson. At Louie’s Bistro, there were 2 salads, 4 entrées, and 3 desserts from which to choose a three-course meal. Using the fundamental counting principle, you can calculate the number of possible meals.

 

2 × 4 × 3 = 24 meals

The fundamental counting principle can also be applied in situations where there are some restrictions on the choices. For example, if you want to determine the number of possible phone numbers, you need to know that you cannot use a zero as the first digit.

 

For many questions, there is a much more efficient way of calculating outcomes than drawing a tree diagram or creating an outcome table. Sometimes it is easier to represent a problem by drawing blanks for the outcomes. For example, how many licence plates can be made with three letters (L) followed by three digits (D)?

 

	Draw blanks to represent the three letters (L) and the three digits (D) in a licence plate.
	There are 26 letters in the alphabet, so each of the first three blanks can be filled in 26 ways (A to Z), assuming you are allowed to repeat letters.
	Each of the digit blanks can be filled in 10 ways (0 to 9).
26 × 26 × 26 × 10 × 10 × 10 = 17 576 000	Therefore, according to the fundamental counting principle, all of the blanks can be filled in 26 × 26 × 26 × 10 × 10 × 10 or 17 576 000 ways.

 

In the next Try This, you will use blanks to help solve a problem.

 

Try This 3

 

A group of 10 children is lined up to go outside for recess. How many different ways can the children line up?

 

Draw 10 blanks to represent the 10 places in line.

 

_ _ _ _ _ _ _ _ _ _

1. How many possible children could be placed in the first spot in line?
2. Now that one child has been placed first in line, how many children are left that could be placed in the second spot?
3. Finish determining the number of possibilities for each place in line.
4. Use the fundamental counting principle to calculate the total number of ways the children can be lined up.

Save your responses in your course folder.

 

Self-Check 2

1. Consider the question about licence plates posed in the Focus section at the start of this lesson. In 2010, Ontario switched from three digits and three letters to four letters and three digits. With a partner or in a group, determine how many more licence plates are available in Ontario now with the fourth letter added. Answer

In Try This 3, you found that you could calculate the number of ways 10 children could be lined up by finding the product of all consecutive natural numbers less than or equal to 10.

 

In mathematics, there is a short-cut notation for the product of a positive integer, n, and all integers less than or equal to n called factorial notation. It is denoted as n!.

 

Therefore, in Try This 3, students can be lined up in 10! ways.

 

Try This 4

 

To learn more about how factorial notation works, use the interactive piece titled Number of Arrangements to line up 10 students.

 

 

Self-Check 3

1. What is the value of 5!? Answer
2. How can 6 × 5 × 4 × 3 × 2 × 1 be written using factorial notation? Answer

Connect

 

Lesson 1 Assignment

Lesson 1 Summary

 

In this lesson you learned how to determine the total number of outcomes in a sample space by using tree diagrams, outcome charts, and the fundamental counting principle. You also learned a short-cut notation called factorial notation. By using both the fundamental counting principle and factorial notation, you have seen that there are many ways to solve problems that involve arranging items and finding the possible number of combinations.

 

In Lesson 2 you will investigate and learn how factorial notation can be used to solve more complex problems.