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![Page 1: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/1.jpg)
Section 7-3Multiplication Counting Principles
![Page 2: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/2.jpg)
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.
![Page 3: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/3.jpg)
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.
![Page 4: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/4.jpg)
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.
![Page 5: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/5.jpg)
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.
![Page 6: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/6.jpg)
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.
![Page 7: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/7.jpg)
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.
![Page 8: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/8.jpg)
Multiplication Counting Principle
![Page 9: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/9.jpg)
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.
![Page 10: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/10.jpg)
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?
![Page 11: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/11.jpg)
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.
![Page 12: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/12.jpg)
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:
![Page 13: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/13.jpg)
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.
![Page 14: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/14.jpg)
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:
![Page 15: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/15.jpg)
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
![Page 16: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/16.jpg)
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%
![Page 17: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/17.jpg)
Arrangement:
![Page 18: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/18.jpg)
Any of the different possible ways that elements of sets can be placed in an order
Arrangement:
![Page 19: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/19.jpg)
Any of the different possible ways that elements of sets can be placed in an order
Arrangement:
Theorem (Selections with Replacement):
![Page 20: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/20.jpg)
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
![Page 21: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/21.jpg)
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?
![Page 22: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/22.jpg)
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
![Page 23: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/23.jpg)
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
![Page 24: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/24.jpg)
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?
![Page 25: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/25.jpg)
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?
![Page 26: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/26.jpg)
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
![Page 27: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/27.jpg)
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?
![Page 28: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/28.jpg)
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
![Page 29: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/29.jpg)
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
![Page 30: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/30.jpg)
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
![Page 31: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/31.jpg)
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
![Page 32: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/32.jpg)
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
![Page 33: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/33.jpg)
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
![Page 34: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/34.jpg)
Example 3How many orders of finish are possible in a 7-horse race?
![Page 35: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/35.jpg)
Example 3How many orders of finish are possible in a 7-horse race?
How many horses can with first place?
![Page 36: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/36.jpg)
Example 3How many orders of finish are possible in a 7-horse race?
How many horses can with first place?7
![Page 37: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/37.jpg)
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.
![Page 38: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/38.jpg)
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.
![Page 39: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/39.jpg)
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?
![Page 40: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/40.jpg)
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
![Page 41: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/41.jpg)
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
![Page 42: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/42.jpg)
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
![Page 43: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/43.jpg)
n Factorial
![Page 44: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/44.jpg)
n FactorialWhen n is a positive real number, n factorial is the product of that number and all positive integers less than that number
![Page 45: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/45.jpg)
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!
![Page 46: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/46.jpg)
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!
![Page 47: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/47.jpg)
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
![Page 48: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/48.jpg)
Theorem (Selections without Replacement)
![Page 49: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/49.jpg)
Theorem (Selections without Replacement)
For any set S with n elements, there are n! possible arrangements of the elements without replacement
![Page 50: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/50.jpg)
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?
![Page 51: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/51.jpg)
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
![Page 52: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/52.jpg)
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
![Page 53: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/53.jpg)
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
![Page 54: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/54.jpg)
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
![Page 55: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/55.jpg)
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( ) !
![Page 56: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/56.jpg)
Homework
![Page 57: Notes 7-3](https://reader033.fdocuments.net/reader033/viewer/2022042817/559c937f1a28aba6128b45b5/html5/thumbnails/57.jpg)
Homework
p. 443 #1-21