Exam 8: Advanced Counting Techniques

Suppose that $| A | = | B | = | C | = 100 , | A \cap B | = 60 , | A \cap C | = 50 , | B \cap C | = 40 , \text { and } | A \cup B \cup C | = 175$ How many elements are in $A \cap B \cap C ?$

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Question 1

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By the principle of inclusion-exclusion $| A \cup B \cup C | = | A | + | B | + | C | - | A \cap B | - | A \cap C | - | B \cap C | + | A \cap B \cap C |$ .
Hence 175=100+100+100-60-50-40+ $| A \cap B \cap C |$ . Therefore $| A \cap B \cap C |$ =175-150=25 .

How many positive integers not exceeding 1000 are not divisible by either 4 or 6?

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Question 2

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The number of positive integers not exceeding 1000 that are not divisible by either 4 or 6 equals 1000 $\lfloor 1000 / 4 \rfloor - \lfloor 1000 / 6 \rfloor + \lfloor 1000 / 12 \rfloor = 1000 - 250 - 166 + 83 = 667$ . Here we used the fact that the integers divisible by both 4 and 6 are those divisible by 12 .

Find the solution of the recurrence relation $a _ { n } = 3 a _ { n - 1 } , \text { with } a _ { 0 } = 2$

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Question 3

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By iteration we find that $a _ { n } = 3 a _ { n - 1 } = 3 \left( 3 a _ { n - 2 } \right) = 3 ^ { 2 } a _ { n - 2 } = \cdots = 3 ^ { n } a _ { 0 } = 2 \cdot 3 ^ { n }$ . This can be verified using mathematical induction.

List the derangements of the set {1, 2, 3, 4}.

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Question 4

How many ways are there to assign six jobs to four employees so that every employee is assigned at least one job?

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Question 5

How many onto functions are there from a set with six elements to a set with four elements?

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Question 6

Find the solution of the linear homogeneous recurrence relation $a _ { n } = 7 a _ { n - 1 } - 6 a _ { n - 2 } \text { with } a _ { 0 } = - 1 \text { and }$ a1 = 4.

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Question 7

How many positive integers not exceeding 1000 are not divisible by 4, 6, or 9?

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Question 8

Use generating functions to solve the recurrence relation $a _ { k } = 5 a _ { k - 1 }$ =1,2,3, ... , with initial condition a₀=3

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Question 9

Find a recurrence relation and initial condition for the number of fruit ﬂies in a jar if there are 12 ﬂies initially and every week there are six times as many ﬂies in the jar as there were the previous week.

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Question 10

Suppose that f(n) satisﬁes the divide-and-conquer relation $f ( n ) = 2 f ( n / 3 ) + 5 \text { and } f ( 1 ) = 7$ What is f(81)?

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Question 11

Suppose that $f ( n )$ satisfies the divide-and-conquer recurrence relation $f ( n ) = 3 f ( n / 4 ) + n ^ { 2 } / 8 \text { with } f ( 1 ) = 2$ What is $f ( 64 )$ ?

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Question 12

How many permutations are there of the digits in the string 12345 that leave 3 ﬁxed but leave no other integer ﬁxed? (For instance, 24351 is such a permutation.)

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Question 13

(a) Find a recurrence relation for the number of ways to climb n stairs if stairs can be climbed two or three at a time. (b) What are the initial conditions? (c) How many ways are there to climb eight stairs?

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Question 14

What is the solution to the recurrence relation $a _ { n } = 8 a _ { n - 1 } + 9 a _ { n - 2 } \text { if } a _ { 0 } = 3 \text { and } a _ { 1 } = 7$

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Question 15

Find a generating function for the sequence 2, 3, 4, 5, . . ..

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Question 16

Exam 8: Advanced Counting Techniques

How many positive integers not exceeding 1000 are not divisible by either 4 or 6?

Find the solution of the recurrence relation an=3an−1, with a0=2a _ { n } = 3 a _ { n - 1 } , \text { with } a _ { 0 } = 2an​=3an−1​, with a0​=2

List the derangements of the set {1, 2, 3, 4}.

How many ways are there to assign six jobs to four employees so that every employee is assigned at least one job?

How many onto functions are there from a set with six elements to a set with four elements?

Find the solution of the linear homogeneous recurrence relation an=7an−1−6an−2 with a0=−1 and a _ { n } = 7 a _ { n - 1 } - 6 a _ { n - 2 } \text { with } a _ { 0 } = - 1 \text { and }an​=7an−1​−6an−2​ with a0​=−1 and a1 = 4.

How many positive integers not exceeding 1000 are not divisible by 4, 6, or 9?

Use generating functions to solve the recurrence relation ak=5ak−1a _ { k } = 5 a _ { k - 1 }ak​=5ak−1​ =1,2,3, ... , with initial condition a0=3

Find a recurrence relation and initial condition for the number of fruit ﬂies in a jar if there are 12 ﬂies initially and every week there are six times as many ﬂies in the jar as there were the previous week.

Suppose that f(n) satisﬁes the divide-and-conquer relation f(n)=2f(n/3)+5 and f(1)=7f ( n ) = 2 f ( n / 3 ) + 5 \text { and } f ( 1 ) = 7f(n)=2f(n/3)+5 and f(1)=7 What is f(81)?

Suppose that f(n)f ( n )f(n) satisfies the divide-and-conquer recurrence relation f(n)=3f(n/4)+n2/8 with f(1)=2f ( n ) = 3 f ( n / 4 ) + n ^ { 2 } / 8 \text { with } f ( 1 ) = 2f(n)=3f(n/4)+n2/8 with f(1)=2 What is f(64)f ( 64 )f(64) ?

How many permutations are there of the digits in the string 12345 that leave 3 ﬁxed but leave no other integer ﬁxed? (For instance, 24351 is such a permutation.)

(a) Find a recurrence relation for the number of ways to climb n stairs if stairs can be climbed two or three at a time. (b) What are the initial conditions? (c) How many ways are there to climb eight stairs?

What is the solution to the recurrence relation an=8an−1+9an−2 if a0=3 and a1=7a _ { n } = 8 a _ { n - 1 } + 9 a _ { n - 2 } \text { if } a _ { 0 } = 3 \text { and } a _ { 1 } = 7an​=8an−1​+9an−2​ if a0​=3 and a1​=7

Find a generating function for the sequence 2, 3, 4, 5, . . ..

The Foundations: Logic and Proofs

A: the Foundations: Logic and Proofs

Basic Structures: Sets, Functions, Sequences, Sums, Matrices

A: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Algorithms

A: Algorithms

Number Theory and Cryptography

A: Number Theory and Cryptography

Induction and Recursion

A: Induction and Recursion

Counting

A: Counting

Discrete Probability

A: Discrete Probability

A: Advanced Counting Techniques

Relations

A: Relations

Graphs

A: Graphs

Trees

A: Trees

Boolean Algebra

A: Boolean Algebra

Modeling Computation

A: Modeling Computation

Mathematics Problem Set: Set Theory, Number Theory, Combinatorics, and Boolean Algebra

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Find the solution of the recurrence relation $a _ { n } = 3 a _ { n - 1 } , \text { with } a _ { 0 } = 2$

Find the solution of the linear homogeneous recurrence relation $a _ { n } = 7 a _ { n - 1 } - 6 a _ { n - 2 } \text { with } a _ { 0 } = - 1 \text { and }$ a1 = 4.

Use generating functions to solve the recurrence relation $a _ { k } = 5 a _ { k - 1 }$ =1,2,3, ... , with initial condition a₀=3

Suppose that f(n) satisﬁes the divide-and-conquer relation $f ( n ) = 2 f ( n / 3 ) + 5 \text { and } f ( 1 ) = 7$ What is f(81)?

Suppose that $f ( n )$ satisfies the divide-and-conquer recurrence relation $f ( n ) = 3 f ( n / 4 ) + n ^ { 2 } / 8 \text { with } f ( 1 ) = 2$ What is $f ( 64 )$ ?

What is the solution to the recurrence relation $a _ { n } = 8 a _ { n - 1 } + 9 a _ { n - 2 } \text { if } a _ { 0 } = 3 \text { and } a _ { 1 } = 7$