Exam 8: Advanced Counting Techniques

In questions ﬁnd a closed form for the generating function for the sequence. - $1 , - 1 , \frac { 1 } { 2 ! } , - \frac { 1 } { 3 ! } , \frac { 1 } { 4 ! } , - \frac { 1 } { 5 ! } , \ldots$

(Short Answer)

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Question 21

In questions ﬁnd a closed form for the generating function for the sequence. -2, 3, 4, 5, 6, 7, . . . .

(Short Answer)

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Question 22

In questions ﬁnd a closed form for the generating function for the sequence. - $\left( \begin{array} { l } 50 \\50\end{array} \right) , \left( \begin{array} { l } 50 \\49\end{array} \right) , \left( \begin{array} { l } 50 \\48\end{array} \right) , \ldots , \left( \begin{array} { c } 50 \\1\end{array} \right) , \left( \begin{array} { c } 50 \\0\end{array} \right) , 0,0,0 , \ldots$

(Short Answer)

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Question 23

A doughnut shop sells 20 kinds of doughnuts. You want to buy 30 doughnuts. How many possibilities are there if you want at most 12 of any one kind?

(Short Answer)

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Question 24

How many permutations of all 26 letters of the alphabet are there that contain at least one of the words DOG, BIG, OIL?

(Short Answer)

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Question 25

In questions ﬁnd a closed form for the generating function for the sequence. -4, 8, 16, 32, 64, . . . .

(Short Answer)

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Question 26

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

(Short Answer)

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Question 27

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 $F ( n ) = \left( n ^ { 2 } + 1 \right) 2 ^ { n }$

(Short Answer)

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Question 28

In questions write the ﬁrst seven terms of the sequence determined by the generating function. - $e ^ { x } + e ^ { - x }$

(Short Answer)

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Question 29

Suppose $f ( n ) = 4 f ( n / 2 ) + n + 2 , f ( 1 ) = 2$ Find $f ( 8 )$ .

(Short Answer)

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Question 30

Set up a generating function and use it to ﬁnd the number of ways in which eleven identical coins can be put in three distinct envelopes if each envelope has at least two coins in it.

(Short Answer)

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Question 31

Set up a generating function and use it to ﬁnd the number of ways in which eleven identical coins can be put in three distinct envelopes if each envelope has an even number of coins in it.

(Short Answer)

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Question 32

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

(Short Answer)

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Question 33

determine whether the recurrence relation is a linear homogeneous recurrence relation with constant coeﬃcients. - $a _ { n } = 5 a _ { n - 1 } ^ { 2 } - 3 a _ { n - 2 } ^ { 2 }$

(Short Answer)

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Question 34

Find the number of bit strings of length eight that contain a pair of consecutive 0's.

(Short Answer)

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Question 35

Assume that the characteristic equation for a homogeneous linear recurrence relation with constant coeﬃcients $\text { is } ( r + 1 ) ^ { 4 } ( r - 1 ) ^ { 4 } = 0 \text {. Describe the form for the general solution to the recurrence relation. }$

(Short Answer)

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Question 36

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 $0,0,0 , a _ { 0 } , a _ { 1 } , a _ { 2 } , \ldots$

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Question 37

Consider the recurrence relation $a _ { n } = 2 a _ { n - 1 } + 3 n$ (a) Write the associated homogeneous recurrence relation. (b) Find the general solution to the associated homogeneous recurrence relation. (c) Find a particular solution to the given recurrence relation. (d) Write the general solution to the given recurrence relation. (e) Find the particular solution to the given recurrence relation when a0 = 1.

(Short Answer)

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Question 38

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 the oldest child gets either 2 or 3 blocks.

(Short Answer)

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Question 39

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

(Short Answer)

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Question 40

In questions ﬁnd a closed form for the generating function for the sequence. - $1 , - 1 , \frac { 1 } { 2 ! } , - \frac { 1 } { 3 ! } , \frac { 1 } { 4 ! } , - \frac { 1 } { 5 ! } , \ldots$

In questions ﬁnd a closed form for the generating function for the sequence. -2, 3, 4, 5, 6, 7, . . . .

A doughnut shop sells 20 kinds of doughnuts. You want to buy 30 doughnuts. How many possibilities are there if you want at most 12 of any one kind?

How many permutations of all 26 letters of the alphabet are there that contain at least one of the words DOG, BIG, OIL?

In questions ﬁnd a closed form for the generating function for the sequence. -4, 8, 16, 32, 64, . . . .

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

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 $F ( n ) = \left( n ^ { 2 } + 1 \right) 2 ^ { n }$

In questions write the ﬁrst seven terms of the sequence determined by the generating function. - $e ^ { x } + e ^ { - x }$

Suppose $f ( n ) = 4 f ( n / 2 ) + n + 2 , f ( 1 ) = 2$ Find $f ( 8 )$ .

Set up a generating function and use it to ﬁnd the number of ways in which eleven identical coins can be put in three distinct envelopes if each envelope has at least two coins in it.

Set up a generating function and use it to ﬁnd the number of ways in which eleven identical coins can be put in three distinct envelopes if each envelope has an even number of coins in it.

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

determine whether the recurrence relation is a linear homogeneous recurrence relation with constant coeﬃcients. - $a _ { n } = 5 a _ { n - 1 } ^ { 2 } - 3 a _ { n - 2 } ^ { 2 }$

Find the number of bit strings of length eight that contain a pair of consecutive 0's.

Assume that the characteristic equation for a homogeneous linear recurrence relation with constant coeﬃcients $\text { is } ( r + 1 ) ^ { 4 } ( r - 1 ) ^ { 4 } = 0 \text {. Describe the form for the general solution to the recurrence relation. }$

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 $0,0,0 , a _ { 0 } , a _ { 1 } , a _ { 2 } , \ldots$

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 the oldest child gets either 2 or 3 blocks.

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

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

Exam 8: Advanced Counting Techniques

In questions ﬁnd a closed form for the generating function for the sequence. - 1,−1,12!,−13!,14!,−15!,…1 , - 1 , \frac { 1 } { 2 ! } , - \frac { 1 } { 3 ! } , \frac { 1 } { 4 ! } , - \frac { 1 } { 5 ! } , \ldots1,−1,2!1​,−3!1​,4!1​,−5!1​,…

In questions ﬁnd a closed form for the generating function for the sequence. -2, 3, 4, 5, 6, 7, . . . .

A doughnut shop sells 20 kinds of doughnuts. You want to buy 30 doughnuts. How many possibilities are there if you want at most 12 of any one kind?

How many permutations of all 26 letters of the alphabet are there that contain at least one of the words DOG, BIG, OIL?

In questions ﬁnd a closed form for the generating function for the sequence. -4, 8, 16, 32, 64, . . . .

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

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+1)2nF ( n ) = \left( n ^ { 2 } + 1 \right) 2 ^ { n }F(n)=(n2+1)2n

In questions write the ﬁrst seven terms of the sequence determined by the generating function. - ex+e−xe ^ { x } + e ^ { - x }ex+e−x

Suppose f(n)=4f(n/2)+n+2,f(1)=2f ( n ) = 4 f ( n / 2 ) + n + 2 , f ( 1 ) = 2f(n)=4f(n/2)+n+2,f(1)=2 Find f(8)f ( 8 )f(8) .

Set up a generating function and use it to ﬁnd the number of ways in which eleven identical coins can be put in three distinct envelopes if each envelope has at least two coins in it.

Set up a generating function and use it to ﬁnd the number of ways in which eleven identical coins can be put in three distinct envelopes if each envelope has an even number of coins in it.

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

determine whether the recurrence relation is a linear homogeneous recurrence relation with constant coeﬃcients. - an=5an−12−3an−22a _ { n } = 5 a _ { n - 1 } ^ { 2 } - 3 a _ { n - 2 } ^ { 2 }an​=5an−12​−3an−22​

Find the number of bit strings of length eight that contain a pair of consecutive 0's.

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 the oldest child gets either 2 or 3 blocks.

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

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 ﬁnd a closed form for the generating function for the sequence. - $1 , - 1 , \frac { 1 } { 2 ! } , - \frac { 1 } { 3 ! } , \frac { 1 } { 4 ! } , - \frac { 1 } { 5 ! } , \ldots$

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

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 $F ( n ) = \left( n ^ { 2 } + 1 \right) 2 ^ { n }$

In questions write the ﬁrst seven terms of the sequence determined by the generating function. - $e ^ { x } + e ^ { - x }$

Suppose $f ( n ) = 4 f ( n / 2 ) + n + 2 , f ( 1 ) = 2$ Find $f ( 8 )$ .

determine whether the recurrence relation is a linear homogeneous recurrence relation with constant coeﬃcients. - $a _ { n } = 5 a _ { n - 1 } ^ { 2 } - 3 a _ { n - 2 } ^ { 2 }$

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