Section 7-3Multiplication Counting Principles

Warm-up1. How many sets of answers are possible if a true-false test has:

a. 1 question? b. 2 questions?

c. 4 questions? d. 10 questions?

2. State a generalization about the number of sets of answers that are possible for a true-false test.

Warm-up1. How many sets of answers are possible if a true-false test has:

a. 1 question? b. 2 questions?

c. 4 questions? d. 10 questions?

2

2. State a generalization about the number of sets of answers that are possible for a true-false test.

Warm-up1. How many sets of answers are possible if a true-false test has:

a. 1 question? b. 2 questions?

c. 4 questions? d. 10 questions?

2 4

2. State a generalization about the number of sets of answers that are possible for a true-false test.

Warm-up1. How many sets of answers are possible if a true-false test has:

a. 1 question? b. 2 questions?

c. 4 questions? d. 10 questions?

2 4

16

2. State a generalization about the number of sets of answers that are possible for a true-false test.

Warm-up1. How many sets of answers are possible if a true-false test has:

a. 1 question? b. 2 questions?

c. 4 questions? d. 10 questions?

2 4

16 1024

2. State a generalization about the number of sets of answers that are possible for a true-false test.

Warm-up1. How many sets of answers are possible if a true-false test has:

a. 1 question? b. 2 questions?

c. 4 questions? d. 10 questions?

2 4

16 1024

2. State a generalization about the number of sets of answers that are possible for a true-false test.

If there are n questions, there are 2n possible sets of answers.

Multiplication Counting Principle

Multiplication Counting Principle

If you have two finite sets, the number of ways to choose something from one set then another is to multiply the sample

spaces of the two events.

Example 1A quiz has 10 questions, each of which can be answered

“always,” “sometimes,” or “never.” If you guess on each question, what is the probability of answering all questions correctly?

Example 1A quiz has 10 questions, each of which can be answered

“always,” “sometimes,” or “never.” If you guess on each question, what is the probability of answering all questions correctly?We need to know the number of favorable outcomes and the

total number of outcomes.

Example 1A quiz has 10 questions, each of which can be answered

“always,” “sometimes,” or “never.” If you guess on each question, what is the probability of answering all questions correctly?We need to know the number of favorable outcomes and the

total number of outcomes.Favorable outcomes:

Example 1A quiz has 10 questions, each of which can be answered

“always,” “sometimes,” or “never.” If you guess on each question, what is the probability of answering all questions correctly?We need to know the number of favorable outcomes and the

total number of outcomes.Favorable outcomes:

Only 1 way to get all of the answers correct.

Example 1A quiz has 10 questions, each of which can be answered

“always,” “sometimes,” or “never.” If you guess on each question, what is the probability of answering all questions correctly?We need to know the number of favorable outcomes and the

total number of outcomes.Favorable outcomes:

Only 1 way to get all of the answers correct.All possible outcomes:

Example 1A quiz has 10 questions, each of which can be answered

“always,” “sometimes,” or “never.” If you guess on each question, what is the probability of answering all questions correctly?We need to know the number of favorable outcomes and the

total number of outcomes.Favorable outcomes:

Only 1 way to get all of the answers correct.All possible outcomes:

310

Example 1A quiz has 10 questions, each of which can be answered

“always,” “sometimes,” or “never.” If you guess on each question, what is the probability of answering all questions correctly?We need to know the number of favorable outcomes and the

total number of outcomes.Favorable outcomes:

Only 1 way to get all of the answers correct.All possible outcomes:

310

The probability of guessing all of the numbers correct is 1/310, or about .00001693508781, or about .001693508781%

Arrangement:

Any of the different possible ways that elements of sets can be placed in an order

Arrangement:

Any of the different possible ways that elements of sets can be placed in an order

Arrangement:

Theorem (Selections with Replacement):

Any of the different possible ways that elements of sets can be placed in an order

Arrangement:

Theorem (Selections with Replacement):For any set S with n elements, there are nk possible arrangements of k elements from S with replacement

Any of the different possible ways that elements of sets can be placed in an order

Arrangement:

Theorem (Selections with Replacement):For any set S with n elements, there are nk possible arrangements of k elements from S with replacement

So what does that mean?

Any of the different possible ways that elements of sets can be placed in an order

Arrangement:

Theorem (Selections with Replacement):For any set S with n elements, there are nk possible arrangements of k elements from S with replacement

So what does that mean?

Draw 3 cards out of a deck and replace after each draw

Any of the different possible ways that elements of sets can be placed in an order

Arrangement:

Theorem (Selections with Replacement):For any set S with n elements, there are nk possible arrangements of k elements from S with replacement

So what does that mean?

Draw 3 cards out of a deck and replace after each draw523 = 140608 possible ways to draw 3 cards

Example 2How many license plates are possible with 4 letters, utilizing the letters A to Z (excluding I and O), followed by 2 digits from

0 to 9?

Example 2How many license plates are possible with 4 letters, utilizing the letters A to Z (excluding I and O), followed by 2 digits from

0 to 9?

How many total letters?

Example 2How many license plates are possible with 4 letters, utilizing the letters A to Z (excluding I and O), followed by 2 digits from

0 to 9?

How many total letters?24

Example 2How many license plates are possible with 4 letters, utilizing the letters A to Z (excluding I and O), followed by 2 digits from

0 to 9?

How many total letters?24

How many total numbers?

Example 2How many license plates are possible with 4 letters, utilizing the letters A to Z (excluding I and O), followed by 2 digits from

0 to 9?

How many total letters?24

How many total numbers?10

Example 2How many license plates are possible with 4 letters, utilizing the letters A to Z (excluding I and O), followed by 2 digits from

0 to 9?

How many total letters?24

How many total numbers?10

244

Example 2How many license plates are possible with 4 letters, utilizing the letters A to Z (excluding I and O), followed by 2 digits from

0 to 9?

How many total letters?24

How many total numbers?10

244 102

Example 2How many license plates are possible with 4 letters, utilizing the letters A to Z (excluding I and O), followed by 2 digits from

0 to 9?

How many total letters?24

How many total numbers?10

244 102

331776

Example 2How many license plates are possible with 4 letters, utilizing the letters A to Z (excluding I and O), followed by 2 digits from

0 to 9?

How many total letters?24

How many total numbers?10

244 102

331776 100

Example 2How many license plates are possible with 4 letters, utilizing the letters A to Z (excluding I and O), followed by 2 digits from

0 to 9?

How many total letters?24

How many total numbers?10

244 102

331776 100

33,177,600 possible plates

Example 3How many orders of finish are possible in a 7-horse race?

Example 3How many orders of finish are possible in a 7-horse race?

How many horses can with first place?

Example 3How many orders of finish are possible in a 7-horse race?

How many horses can with first place?7

Example 3How many orders of finish are possible in a 7-horse race?

How many horses can with first place?7

How many are left for second? Third? Etc.

Example 3How many orders of finish are possible in a 7-horse race?

How many horses can with first place?7

How many are left for second? Third? Etc.6, 5, etc.

Example 3How many orders of finish are possible in a 7-horse race?

How many horses can with first place?7

How many are left for second? Third? Etc.6, 5, etc.

So, what do we get for the possible orders?

Example 3How many orders of finish are possible in a 7-horse race?

How many horses can with first place?7

How many are left for second? Third? Etc.6, 5, etc.

So, what do we get for the possible orders?

7i6i5i4i3i2i1

Example 3How many orders of finish are possible in a 7-horse race?

How many horses can with first place?7

How many are left for second? Third? Etc.6, 5, etc.

So, what do we get for the possible orders?

7i6i5i4i3i2i1

= 5040 possible orders

Example 3How many orders of finish are possible in a 7-horse race?

How many horses can with first place?7

How many are left for second? Third? Etc.6, 5, etc.

So, what do we get for the possible orders?

7i6i5i4i3i2i1

This is an idea known as a factorial.

= 5040 possible orders

n Factorial

n FactorialWhen n is a positive real number, n factorial is the product of that number and all positive integers less than that number

n FactorialWhen n is a positive real number, n factorial is the product of that number and all positive integers less than that number

Notation: n!

n FactorialWhen n is a positive real number, n factorial is the product of that number and all positive integers less than that number

Notation: n!

In Example 3, we saw 7!

n FactorialWhen n is a positive real number, n factorial is the product of that number and all positive integers less than that number

Notation: n!

In Example 3, we saw 7!

7! = 7i6i5i4i3i2i1

Theorem (Selections without Replacement)

Theorem (Selections without Replacement)

For any set S with n elements, there are n! possible arrangements of the elements without replacement

Theorem (Selections without Replacement)

For any set S with n elements, there are n! possible arrangements of the elements without replacement

So what does that mean?

Theorem (Selections without Replacement)

For any set S with n elements, there are n! possible arrangements of the elements without replacement

So what does that mean?

Draw 3 cards out of a deck and don’t replace after each draw

Theorem (Selections without Replacement)

For any set S with n elements, there are n! possible arrangements of the elements without replacement

So what does that mean?

Draw 3 cards out of a deck and don’t replace after each draw

52i51i50

Theorem (Selections without Replacement)

For any set S with n elements, there are n! possible arrangements of the elements without replacement

So what does that mean?

Draw 3 cards out of a deck and don’t replace after each draw

52i51i50=132600 possible ways

Theorem (Selections without Replacement)

For any set S with n elements, there are n! possible arrangements of the elements without replacement

So what does that mean?

Draw 3 cards out of a deck and don’t replace after each draw

52i51i50=132600 possible ways

We would use 52! if we were drawing every card from the deck....that’s A LOT of possible arrangements

ChallengeSimplify the following and explain how you solved them for

bonus points (Goes into Quiz category, 2 pts each)

1. k +1( ) !− k ! 2.

k +1( ) !− k !k !

3.

k +1( ) !+ k !k +1( ) !− k !

4.1k !

−1

k +1( ) !

Homework

Homework

p. 443 #1-21