Exam 8: Advanced Counting Techniques

In questions , describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. - $a _ { n }$ = the number of bit strings of length n with an even number of 0's.

(Short Answer)

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Question 101

A doughnut shop sells 20 kinds of doughnuts. You want to buy 30 doughnuts. How many possibilities are there if you want at most six of any one kind?

(Essay)

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Question 102

determine whether the recurrence relation is a linear homogeneous recurrence relation with constant coeﬃcients. - $a _ { n } = n a _ { n - 1 }$

(Short Answer)

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Question 103

In questions solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - $a _ { n } = 2 a _ { n - 1 } + 5 , \quad a _ { 0 } = 3$

(Short Answer)

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Question 104

Find the number of strings of 0's, 1's, and 2's of length six that have no consecutive 0's.

(Short Answer)

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Question 105

If $G ( x )$ is the generating function for $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ describe in terms of $G ( x )$ the generating function for $5 , a _ { 1 } , 0 , a _ { 3 } , a _ { 4 } , a _ { 5 } , \ldots$

(Short Answer)

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Question 106

A market sells ten kinds of soda. You want to buy 12 bottles. How many possibilities are there? if you want at most three bottles of any kind?

(Short Answer)

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Question 107

ﬁnd the coeﬃcient of $x ^ { 8 }$ in the power series of each of the function. - $\left( 1 + x ^ { 2 } + x ^ { 4 } \right) ^ { 3 }$

(Short Answer)

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Question 108

In questions write the ﬁrst seven terms of the sequence determined by the generating function. -cos x.

(Short Answer)

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Question 109

In questions solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - $a _ { n } = 3 n a _ { n - 1 } , a _ { 0 } = 2$

(Short Answer)

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Question 110

Assume that the characteristic equation for a homogeneous linear recurrence relation with constant coeﬃcients $\text { is } ( r - 5 ) ^ { 3 } = 0 \text {. }$ Describe the form for the general solution to the recurrence relation.

(Short Answer)

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Question 111

Set up a generating function and use it to ﬁnd the number of ways in which nine identical blocks can be given to four children, if each child gets at least two blocks.

(Short Answer)

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Question 112

Find $\left| A _ { 1 } \cup A _ { 2 } \cup A _ { 3 } \cup A _ { 4 } \right|$ if each set A_i has 150 elements, each intersection of two sets has 80 elements, each intersection of three sets has 20 elements, and there are no elements in all four sets.

(Short Answer)

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Question 113

Use generating functions to solve $a _ { n } = 3 a _ { n - 1 } + 2 ^ { n } , a _ { 0 } = 5$

(Short Answer)

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Question 114

A market sells 40 kinds of candy bars. You want to buy 15 candy bars. (a) How many possibilities are there? (b) How many possibilities are there if you want at least three peanut butter bars and at least ﬁve almond bars? (c) How many possibilities are there if you want exactly three peanut butter bars and exactly ﬁve almond bars? (d) How many possibilities are there if you want at most four toﬀee bars and at most six mint bars?

(Short Answer)

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Question 115

In questions ﬁnd a closed form for the generating function for the sequence. -1, 0, 1, 0, 1, 0, 1, 0, . . . .

(Short Answer)

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Question 116

ﬁnd the coeﬃcient of $x ^ { 8 }$ in the power series of each of the function. - $1 / \left( 1 - 3 x ^ { 2 } \right)$

(Short Answer)

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Question 117

In questions solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - $a _ { n } = 2 a _ { n - 1 } + 2 a _ { n - 2 } , \quad a _ { 0 } = 0 , a _ { 1 } = 1$

(Short Answer)

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Question 118

ﬁnd the coeﬃcient of $x ^ { 8 }$ in the power series of each of the function. - $x ^ { 2 } / ( 1 + 2 x ) ^ { 2 }$

(Short Answer)

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Question 119

What form does a particular solution of the linear nonhomogeneous recurrence relation $a _ { n } = 4 a _ { n - 1 } - 4 a _ { n - 2 }$ $F ( n )$ have when $F ( n ) = n ^ { 2 } \cdot 4 ^ { n }$

(Short Answer)

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Question 120

In questions , describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. - $a _ { n }$ = the number of bit strings of length n with an even number of 0's.

A doughnut shop sells 20 kinds of doughnuts. You want to buy 30 doughnuts. How many possibilities are there if you want at most six of any one kind?

determine whether the recurrence relation is a linear homogeneous recurrence relation with constant coeﬃcients. - $a _ { n } = n a _ { n - 1 }$

In questions solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - $a _ { n } = 2 a _ { n - 1 } + 5 , \quad a _ { 0 } = 3$

Find the number of strings of 0's, 1's, and 2's of length six that have no consecutive 0's.

If $G ( x )$ is the generating function for $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ describe in terms of $G ( x )$ the generating function for $5 , a _ { 1 } , 0 , a _ { 3 } , a _ { 4 } , a _ { 5 } , \ldots$

A market sells ten kinds of soda. You want to buy 12 bottles. How many possibilities are there? if you want at most three bottles of any kind?

ﬁnd the coeﬃcient of $x ^ { 8 }$ in the power series of each of the function. - $\left( 1 + x ^ { 2 } + x ^ { 4 } \right) ^ { 3 }$

In questions write the ﬁrst seven terms of the sequence determined by the generating function. -cos x.

In questions solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - $a _ { n } = 3 n a _ { n - 1 } , a _ { 0 } = 2$

Assume that the characteristic equation for a homogeneous linear recurrence relation with constant coeﬃcients $\text { is } ( r - 5 ) ^ { 3 } = 0 \text {. }$ Describe the form for the general solution to the recurrence relation.

Set up a generating function and use it to ﬁnd the number of ways in which nine identical blocks can be given to four children, if each child gets at least two blocks.

Find $\left| A _ { 1 } \cup A _ { 2 } \cup A _ { 3 } \cup A _ { 4 } \right|$ if each set A_i has 150 elements, each intersection of two sets has 80 elements, each intersection of three sets has 20 elements, and there are no elements in all four sets.

Use generating functions to solve $a _ { n } = 3 a _ { n - 1 } + 2 ^ { n } , a _ { 0 } = 5$

In questions ﬁnd a closed form for the generating function for the sequence. -1, 0, 1, 0, 1, 0, 1, 0, . . . .

ﬁnd the coeﬃcient of $x ^ { 8 }$ in the power series of each of the function. - $1 / \left( 1 - 3 x ^ { 2 } \right)$

In questions solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - $a _ { n } = 2 a _ { n - 1 } + 2 a _ { n - 2 } , \quad a _ { 0 } = 0 , a _ { 1 } = 1$

ﬁnd the coeﬃcient of $x ^ { 8 }$ in the power series of each of the function. - $x ^ { 2 } / ( 1 + 2 x ) ^ { 2 }$

What form does a particular solution of the linear nonhomogeneous recurrence relation $a _ { n } = 4 a _ { n - 1 } - 4 a _ { n - 2 }$ $F ( n )$ have when $F ( n ) = n ^ { 2 } \cdot 4 ^ { n }$

The Foundations: Logic and Proofs

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Exam 8: Advanced Counting Techniques

In questions , describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. - ana _ { n }an​ = the number of bit strings of length n with an even number of 0's.

A doughnut shop sells 20 kinds of doughnuts. You want to buy 30 doughnuts. How many possibilities are there if you want at most six of any one kind?

determine whether the recurrence relation is a linear homogeneous recurrence relation with constant coeﬃcients. - an=nan−1a _ { n } = n a _ { n - 1 }an​=nan−1​

In questions solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - an=2an−1+5,a0=3a _ { n } = 2 a _ { n - 1 } + 5 , \quad a _ { 0 } = 3an​=2an−1​+5,a0​=3

Find the number of strings of 0's, 1's, and 2's of length six that have no consecutive 0's.

A market sells ten kinds of soda. You want to buy 12 bottles. How many possibilities are there? if you want at most three bottles of any kind?

ﬁnd the coeﬃcient of x8x ^ { 8 }x8 in the power series of each of the function. - (1+x2+x4)3\left( 1 + x ^ { 2 } + x ^ { 4 } \right) ^ { 3 }(1+x2+x4)3

In questions write the ﬁrst seven terms of the sequence determined by the generating function. -cos x.

In questions solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - an=3nan−1,a0=2a _ { n } = 3 n a _ { n - 1 } , a _ { 0 } = 2an​=3nan−1​,a0​=2

Assume that the characteristic equation for a homogeneous linear recurrence relation with constant coeﬃcients is (r−5)3=0. \text { is } ( r - 5 ) ^ { 3 } = 0 \text {. } is (r−5)3=0. Describe the form for the general solution to the recurrence relation.

Set up a generating function and use it to ﬁnd the number of ways in which nine identical blocks can be given to four children, if each child gets at least two blocks.

Use generating functions to solve an=3an−1+2n,a0=5a _ { n } = 3 a _ { n - 1 } + 2 ^ { n } , a _ { 0 } = 5an​=3an−1​+2n,a0​=5

In questions ﬁnd a closed form for the generating function for the sequence. -1, 0, 1, 0, 1, 0, 1, 0, . . . .

ﬁnd the coeﬃcient of x8x ^ { 8 }x8 in the power series of each of the function. - 1/(1−3x2)1 / \left( 1 - 3 x ^ { 2 } \right)1/(1−3x2)

In questions solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - an=2an−1+2an−2,a0=0,a1=1a _ { n } = 2 a _ { n - 1 } + 2 a _ { n - 2 } , \quad a _ { 0 } = 0 , a _ { 1 } = 1an​=2an−1​+2an−2​,a0​=0,a1​=1

ﬁnd the coeﬃcient of x8x ^ { 8 }x8 in the power series of each of the function. - x2/(1+2x)2x ^ { 2 } / ( 1 + 2 x ) ^ { 2 }x2/(1+2x)2

What form does a particular solution of the linear nonhomogeneous recurrence relation an=4an−1−4an−2a _ { n } = 4 a _ { n - 1 } - 4 a _ { n - 2 }an​=4an−1​−4an−2​ F(n)F ( n )F(n) have when F(n)=n2⋅4nF ( n ) = n ^ { 2 } \cdot 4 ^ { n }F(n)=n2⋅4n

The Foundations: Logic and Proofs

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

Algorithms

Number Theory and Cryptography

Induction and Recursion

Counting

Discrete Probability

Relations

Graphs

Trees

Boolean Algebra

Modeling Computation

Filters

In questions , describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. - $a _ { n }$ = the number of bit strings of length n with an even number of 0's.

determine whether the recurrence relation is a linear homogeneous recurrence relation with constant coeﬃcients. - $a _ { n } = n a _ { n - 1 }$

In questions solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - $a _ { n } = 2 a _ { n - 1 } + 5 , \quad a _ { 0 } = 3$

ﬁnd the coeﬃcient of $x ^ { 8 }$ in the power series of each of the function. - $\left( 1 + x ^ { 2 } + x ^ { 4 } \right) ^ { 3 }$

In questions solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - $a _ { n } = 3 n a _ { n - 1 } , a _ { 0 } = 2$

Assume that the characteristic equation for a homogeneous linear recurrence relation with constant coeﬃcients $\text { is } ( r - 5 ) ^ { 3 } = 0 \text {. }$ Describe the form for the general solution to the recurrence relation.

Use generating functions to solve $a _ { n } = 3 a _ { n - 1 } + 2 ^ { n } , a _ { 0 } = 5$

ﬁnd the coeﬃcient of $x ^ { 8 }$ in the power series of each of the function. - $1 / \left( 1 - 3 x ^ { 2 } \right)$

In questions solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - $a _ { n } = 2 a _ { n - 1 } + 2 a _ { n - 2 } , \quad a _ { 0 } = 0 , a _ { 1 } = 1$

ﬁnd the coeﬃcient of $x ^ { 8 }$ in the power series of each of the function. - $x ^ { 2 } / ( 1 + 2 x ) ^ { 2 }$

What form does a particular solution of the linear nonhomogeneous recurrence relation $a _ { n } = 4 a _ { n - 1 } - 4 a _ { n - 2 }$ $F ( n )$ have when $F ( n ) = n ^ { 2 } \cdot 4 ^ { n }$