Notes 7-3
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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