BellworkBellworkRemember that class begins when you enter the Remember that class begins when you enter the

classroom.classroom.

1)1) At a sporting goods store, skateboards are At a sporting goods store, skateboards are available in 8 different deck designs. Each available in 8 different deck designs. Each deck design is available with 4 different wheel deck design is available with 4 different wheel assemblies. How many skateboard choices assemblies. How many skateboard choices does the store offer?does the store offer?

2)2) You would like to go to a movie, a play, or the You would like to go to a movie, a play, or the zoo. You can go with your cousin, your brother, zoo. You can go with your cousin, your brother, or your friend. You can go on Friday, Saturday, or your friend. You can go on Friday, Saturday, or Sunday. How many different options do you or Sunday. How many different options do you have?have?

Solutions:Solutions:1)1) Multiply the number of deck designs (8) Multiply the number of deck designs (8)

and the number of wheel assemblies (4). and the number of wheel assemblies (4). So the store offers 32 skateboard So the store offers 32 skateboard choices.choices.

2)2) Multiply the number of places you can Multiply the number of places you can go to, times the number of different go to, times the number of different people you can go with, times the people you can go with, times the number of different days you can you number of different days you can you on:on:

3 * 3 * 3 = 27 different options.3 * 3 * 3 = 27 different options.

There are two There are two Counting Principles:Counting Principles:

The multiplication counting The multiplication counting principleprinciple

The addition counting principleThe addition counting principle

We’ve already been using the We’ve already been using the multiplication counting principle. multiplication counting principle. Today’s bellwork was an example. Let’s Today’s bellwork was an example. Let’s look at the definition:look at the definition:

Multiplication counting principleMultiplication counting principle:: If If one event can occur in one event can occur in mm ways and ways and another even can occur in another even can occur in nn ways, then ways, then the number of ways that the number of ways that bothboth events events can occur together is m x n. can occur together is m x n.

Multiplication counting principleMultiplication counting principle:: If one event can occur in If one event can occur in mm ways and ways and another even can occur in another even can occur in nn ways, then the number of ways that ways, then the number of ways that bothboth events can occur together is m * n. events can occur together is m * n.

For example: On your history test you can For example: On your history test you can choose one of 4 essay questions and one of 5 choose one of 4 essay questions and one of 5 extra credit questions. How many choices do extra credit questions. How many choices do you have:you have:

The first event is choosing your essay question, The first event is choosing your essay question, and there are 4 choices, so m=4. and there are 4 choices, so m=4.

The second even is choosing your extra credit The second even is choosing your extra credit question, and there are 5 choices, so n=5.question, and there are 5 choices, so n=5.

The total number of choices you have is m*n = The total number of choices you have is m*n = 4*5 = 20.4*5 = 20.

At a shoe store, shoes are available in 6 different styles and 3 different colors. How many choices does the shoe store offer?

Before we define the addition counting principle, Before we define the addition counting principle, let’s look at a few examples:let’s look at a few examples:

Every student at Osborne is given an Every student at Osborne is given an id code consisting of 3 symbols (first id code consisting of 3 symbols (first a letter and then two digits). How a letter and then two digits). How many different id codes are possible?many different id codes are possible?

Every student at Osborne is given an id code consisting of 3 symbols Every student at Osborne is given an id code consisting of 3 symbols (letters and digits). How many different id codes are possible if at least one (letters and digits). How many different id codes are possible if at least one

letter is used in each? letter is used in each?

The code looks like this:The code looks like this:LDDLDD

How many letters are there to choose from?How many letters are there to choose from?2626

How many numbers do we have to choose from for each How many numbers do we have to choose from for each digit?digit?1010

So we have 26 x 10 x 10 different codes.So we have 26 x 10 x 10 different codes.

26 x 10 x 10 = 260026 x 10 x 10 = 2600

What if we need more id What if we need more id codes???codes???

Every student at Osborne is Every student at Osborne is given an id code consisting of given an id code consisting of 3 symbols (letters and digits). 3 symbols (letters and digits). How many different id codes How many different id codes are possible if at least one are possible if at least one letter is used in each?letter is used in each?

Every student at Osborne is given an id code consisting of 3 symbols Every student at Osborne is given an id code consisting of 3 symbols (letters and digits). How many different id codes are possible if at least one (letters and digits). How many different id codes are possible if at least one

letter is used in each? letter is used in each?

To find the number of codes, first To find the number of codes, first determine the number of ways each determine the number of ways each code can be made. We can have code can be made. We can have either a letter (L) or a digit (D). There either a letter (L) or a digit (D). There are 26 different letters and 10 are 26 different letters and 10 different digits (0-9). Each code must different digits (0-9). Each code must have have at least oneat least one letter, so we have: letter, so we have:

LDD, DLD, DDL, LLD, LDL, DLL, LLLLDD, DLD, DDL, LLD, LDL, DLL, LLL

CodeCode Multiplication Counting Multiplication Counting PrinciplePrinciple

LDDLDD 26 * 10 * 10 = 2,60026 * 10 * 10 = 2,600

DLDDLD 10 * 26 * 10 = 2,60010 * 26 * 10 = 2,600

DDLDDL 10 * 10 * 26 = 2,60010 * 10 * 26 = 2,600

LLDLLD 26 * 26 * 10 = 6,76026 * 26 * 10 = 6,760

LDLLDL 26 * 10 * 26 = 6,76026 * 10 * 26 = 6,760

DLLDLL 10 * 26 * 26 = 6,76010 * 26 * 26 = 6,760

LLLLLL 26 * 26 * 26 = 26 * 26 * 26 = 17,576 17,576 ++Addition Counting Principle: 45,656Addition Counting Principle: 45,656

VocabularyVocabulary

Addition counting principleAddition counting principle:: If If the possibilities being counted can the possibilities being counted can be split into different groups, then be split into different groups, then the total number of possibilities is the total number of possibilities is the sum of the numbers of the sum of the numbers of possibilities for each group.possibilities for each group.

Class ExampleClass Example

Let’s pretend that I am getting to Let’s pretend that I am getting to choose 4 students from this class to choose 4 students from this class to be given a free homework pass. I be given a free homework pass. I must choose at least 1 boy and 1 must choose at least 1 boy and 1 girl. Based on the students in this girl. Based on the students in this room, how many different ways can I room, how many different ways can I choose this group?choose this group?