Lesson 1

Lesson 1: Fundamental Counting Principle

Focus

One of the ways people are identified is by phone numbers. In the not-too-distant past, one phone number was often associated with an entire family. Now, however, many people possess individual cellphones and their own unique phone numbers.

Prior to 1999, the area code for the entire province of Alberta was 403. In 1999, the southern half of the province retained the 403 area code, and the northern half was assigned the new area code of 780. In September 2008, Alberta added another area code, 587, across the province as available phone numbers were again running out. By adding another area code there is a potential for nine million more phone numbers.

Can you think why people in Alberta continue to need more phone numbers? By the end of this lesson you will be able to show that a new area code does indeed amount to nine million new phone numbers.

Determining the number of ways things may occur is a branch of mathematics called combinatorics. This branch is used in many ways: from designing a computer network to coding your 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 the number 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 30-1 LearnEveryware 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

If your Personal Identification Number (PIN) consists of four digits, how long do you think it would take a thief to hack your number? In theory there are 10 000 possible PINs available but, in reality, most banks have restrictions on what digits may be used. Most bank machines only allow three attempts to get the correct PIN, so there is only a 0.06% chance of a thief getting the correct number before the machine locks the crook out.

Try This 1

Abdy received a new chip-enabled credit card. In order to activate the card, Abdy has to choose a PIN. Abdy has several PINs: one for his bank card, one to unlock his telephone, and some for his other credit cards. To make it easier to remember all his pass codes, Abdy always uses an arrangement of his birth date for his PIN. Abdy was born on January 7, 1996. Because of his birth date, Abdy always uses a 1 or a 7 for the first digit, a 9 or a 6 for the second digit, a 1 or a 7 for the third digit, and, finally, a 9 or a 6 for the last digit.

Using Abdy’s restrictions, how many different four-digit PINs are possible?

Save your responses in your course folder.

Share 1

With a partner or in a group, discuss the following.

1. How many PINs are possible?
   
2. Was there a more efficient way used by one of the group members for determining the number of PINs rather than writing them all out?
   
3. Apply the most efficient method you discussed to show how 10 000 possible four-digit PINs can be created when any digit from 0 to 9 can be used in any order.

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

Explore

In Share 1, one of your group members may have suggested using a diagram to solve the PIN problem. The following diagram is called a tree diagram.

By counting the total number of terminal branches in the tree diagrams, you can see that Abdy has a choice of 16 different PINs. The list of all 16 possible PINs is called the sample space for this problem. The sample space is a list of all possible outcomes. You could also list all possible outcomes in an outcome table, such as the following one.

Read through Mathematics Glossary Tree Diagram to see how all possible outcomes of tossing a penny and rolling a die can be calculated by using a tree diagram. As you read, note that the order of tossing the penny or rolling the die does not affect the outcomes or sample space. Read the first example only in Mathematics Glossary Tree Diagram.

In Try This 2 you will find possible outcomes using tree diagrams and outcome tables. You will then determine an expression to calculate these outcomes.

Try This 2

In question 4 of Try This 2 you were asked to make a conjecture about how you can use multiplication to determine the number of different possible outcomes.

To determine the total number of outfits possible, you will multiply the number of shirts by the number of pants and then multiply by the number of shoes or hats. In question 5 you need this product to be 1000 outfits. One possible solution is to have a choice of 10 shirts, 10 pairs of pants, and 10 pairs of shoes. The total number of outfits possible would be 10 × 10 × 10 = 1000.

For question 3.b. in Try This 2 you used a tree diagram to find the number of different outfits or outcomes. Your tree diagram may have looked similar to the diagram shown here.

This tree identified 16 different possible outfits from 4 pairs of pants, 2 shirts, and 2 hats. You may have noticed that you can also determine the total number of outfits possible by multiplying the number of pairs of pants by shirts and by hats: 4 × 2 × 2 = 16.

Think back to the example from Discover. For Abdy’s PIN, there were two choices for each space in the code. By using the fundamental counting principle, you can calculate the number of possible PINs.

2 × 2 × 2 × 2 = 16 possible PINs

Self-Check 1

1. Roger is choosing an outfit to wear for the day. He has a choice of 5 pairs of pants and 8 different shirts. How many different outfits can he choose? Answer

    

   
   Jupiterimages/Brand X Pictures/Thinkstock
   
   
   
2. Use the following information to find the possible number of menu combinations.
   
   
   The graduation council has been given the task of determining the menu for the graduation banquet. They have to select one salad, one main course, one side dish, one vegetable dish, and one dessert from the following choices.
   
   salad: caesar, garden, tomato-herb, pasta
   main course: chicken, beef, fish
   side dish: rice, baked potato, mashed potato, French fries, pasta
   vegetable: peas, corn, carrots, mixed vegetables, broccoli
   dessert: cheesecake, pudding, ice cream, pie
   
   Answer
   
3. How many routes are there from point A to point B if you always travel from left to right? Answer
   
   

Try This 3

Ten children line up to go on the school bus. In how many ways can the children line up? Draw ten blanks to represent the ten places in line.

_ _ _ _ _ _ _ _ _ _

1. How many possible children could be in the first spot in line?
2. Now that one child has been placed first in line, how many possible children could be 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.

As you realized in Try This 3, the fundamental counting principle can also be applied in situations where there are some choice restrictions. 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 more efficient way of calculating outcomes than drawing a tree diagram or creating an outcome table. In Try This 3 you investigated solving one problem by drawing blanks for the outcomes.

Another example to check out is how many license plates can be made with three letters (L) followed by three digits (D).

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

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. This short cut is called factorial notation and is denoted n!. Using factorial notation, the solution to Try This 3 would be that students can be lined up in 10! ways.

To learn more about how factorial notation works, work through Number of Arrangements. You will determine how many ways ten people can be arranged in the lineup. To begin, you will be asked to state how many people are available to fill the first spot in line.

Image adapted from
© Peter Vaclavek
/23581614/Fotolia

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

You have seen that 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1.

If you let n = 10, then the expression can be written as the following:

n! = n × (n − 1) × (n − 2) × (n − 3) × (n − 4 ) × ( n − 5) × ( n − 6) × ( n − 7) × ( n − 8) × ( n − 9)

or

n! = n(n − 1)(n − 2)(n − 3)…(3)(2)(1)

Calculating the value of an expression involving factorial notation without a calculator can be time consuming. In Try This 4 you will look for patterns that might exist between subsequent expressions involving factorial notation.

Try This 4

1. Complete the following table based on the information already available.
   
   

   n!	n! in Expanded Form	Expression Involving n and (n − 1)!	Value of n!
   1!	1	1	1
   2!	2 × 1	2 × _!	2
   3!	3 × 2 × 1	3 × _!
   4!	_ × _ × _ × _	4 × _!
   5!	_ × _ × _ × _ × _	5 × _!
   6!	_ × _ × _ × _ × _ × _	6 × _!
   7!	_ × _ ×  _ × _ × _ × _ × _	7 × _!

2. Describe the relationship between n! and (n − 1)!.

3. Create a general expression relating n!, n, and (n − 1)!.

Save your responses in your course folder.

In Try This 4 you probably noticed that because n! represents the product of consecutive whole numbers, you can write a product in a variety of ways. For example, 10! can also be expressed as 10 x 9! or 10 x 9 x 8!.

Self-Check 4

1. Rewrite 12! in two different forms. Answer
2. Rewrite 7 x 6 x 5! in two different forms. Answer

There are times when it is necessary to simplify factorial expressions. You will explore one way to simplify in Try This 5.

Try This 5

Find the value of

1. Expand the numerator to show all the factors in 12!.
2. Expand the denominator to show all the factors in 9!.
3. Divide the numerator and denominator by any common factors.
4. Calculate the solution by multiplying the remaining factors.

Save your responses in your course folder.

As you may have noticed in Try This 5, writing out all the factors can take a long time. Here is another way you could simplify

Expanding the numerator and denominator gives you

As you can see, the expression in the brackets is the same as 9!. Therefore, you can write as

Dividing the numerator and denominator by the common factor of 9! leaves 12 x 11 x 10, or 1320.

Self-Check 5

1. Simplify.
   1. Answer
   2. 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 arrangements.