Exam 8: A: Advanced Counting Techniques

Find $\left| A _ { 1 } \cup A _ { 2 } \cup A _ { 3 } \cup A _ { 4 } \right| \text { if each set } A _ { i }$ 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 21

ﬁnd the coeﬃcient of x8 in the power series of each of the function. - $\left( 1 + x ^ { 2 } + x ^ { 4 } + x ^ { 6 } + x ^ { 8 } + x ^ { 10 } \right) ^ { 3 }$

(Short Answer)

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Question 22

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?

(Short Answer)

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Question 23

Find the number of ways to put eight diﬀerent books in ﬁve boxes, if no box is allowed to be empty.

(Short Answer)

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Question 24

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 25

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for a₀, 3a₁, 9a₂, 27a₃, 81a₄, . . . .

(Short Answer)

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Question 26

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 27

A market sells ten kinds of soda. You want to buy 12 bottles. How many possibilities are there if you want (a) at least one of each kind? (b) at most seven bottles of any kind?

(Short Answer)

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Question 28

ﬁnd a closed form for the generating function for the sequence. -1, −1, 12!, −3!1 , 14!, − 15!, . . .

(Short Answer)

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Question 29

determine whether the recurrence relation is a linear homogeneous recurrence relation with constant coeﬃcients. - $a _ { n } - 7 a _ { n - 2 } + a _ { n - 5 } = 0$

(Short Answer)

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Question 30

Consider the recurrence relation $a _ { n } = 3 a _ { n - 1 } + 5 ^ { n }$ (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 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 at least two but no more than ﬁve coins in it.

(Short Answer)

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Question 32

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

(Short Answer)

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Question 33

solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - $a _ { n } = - 10 a _ { n - 1 } - 21 a _ { n - 2 } , \quad a _ { 0 } = 2 , a _ { 1 } = 1$

(Short Answer)

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Question 34

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 most ﬁve blocks.

(Short Answer)

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Question 35

Suppose $f ( n ) = 3 f ( n / 2 ) + 1 , f ( 1 ) = 1$ . Find $f ( 8 )$ .

(Short Answer)

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Question 36

Consider the recurrence relation $a_{n}=-a_{n-1}+n$ (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 37

Suppose $f ( n ) = 2 f ( n / 2 ) + 3 , f ( 16 ) = 51$ . Find $\text { Find } f ( 2$ .

(Short Answer)

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Question 38

Suppose $f ( n ) = 2 f ( n / 2 ) , f ( 8 ) = 2$ . Find $\text { Find } f ( 1 )$ .

(Short Answer)

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Question 39

Suppose you have 100 identical marbles and ﬁve jars (labeled A, B, C, D, E). In how many ways can you put the marbles in the jars if: (a) each jar has at least six marbles in it? (b) each jar has at most forty marbles in it?

(Short Answer)

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Question 40

Exam 8: A: Advanced Counting Techniques

ﬁnd the coeﬃcient of x8 in the power series of each of the function. - (1+x2+x4+x6+x8+x10)3\left( 1 + x ^ { 2 } + x ^ { 4 } + x ^ { 6 } + x ^ { 8 } + x ^ { 10 } \right) ^ { 3 }(1+x2+x4+x6+x8+x10)3

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?

Find the number of ways to put eight diﬀerent books in ﬁve boxes, if no box is allowed to be empty.

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​

If G(x) is the generating function for a0, a1, a2, a3, . . . , describe in terms of G(x) the generating function for a0, 3a1, 9a2, 27a3, 81a4, . . . .

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?

A market sells ten kinds of soda. You want to buy 12 bottles. How many possibilities are there if you want (a) at least one of each kind? (b) at most seven bottles of any kind?

ﬁnd a closed form for the generating function for the sequence. -1, −1, 12!, −3!1 , 14!, − 15!, . . .

determine whether the recurrence relation is a linear homogeneous recurrence relation with constant coeﬃcients. - an−7an−2+an−5=0a _ { n } - 7 a _ { n - 2 } + a _ { n - 5 } = 0an​−7an−2​+an−5​=0

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 but no more than ﬁve coins in it.

solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - an=−10an−1−21an−2,a0=2,a1=1a _ { n } = - 10 a _ { n - 1 } - 21 a _ { n - 2 } , \quad a _ { 0 } = 2 , a _ { 1 } = 1an​=−10an−1​−21an−2​,a0​=2,a1​=1

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 most ﬁve blocks.

Suppose f(n)=3f(n/2)+1,f(1)=1f ( n ) = 3 f ( n / 2 ) + 1 , f ( 1 ) = 1f(n)=3f(n/2)+1,f(1)=1 . Find f(8)f ( 8 )f(8) .

Suppose f(n)=2f(n/2)+3,f(16)=51f ( n ) = 2 f ( n / 2 ) + 3 , f ( 16 ) = 51f(n)=2f(n/2)+3,f(16)=51 . Find Find f(2\text { Find } f ( 2 Find f(2 .

Suppose f(n)=2f(n/2),f(8)=2f ( n ) = 2 f ( n / 2 ) , f ( 8 ) = 2f(n)=2f(n/2),f(8)=2 . Find Find f(1)\text { Find } f ( 1 ) Find f(1) .

Suppose you have 100 identical marbles and ﬁve jars (labeled A, B, C, D, E). In how many ways can you put the marbles in the jars if: (a) each jar has at least six marbles in it? (b) each jar has at most forty marbles in it?

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

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

Filters

ﬁnd the coeﬃcient of x8 in the power series of each of the function. - $\left( 1 + x ^ { 2 } + x ^ { 4 } + x ^ { 6 } + x ^ { 8 } + x ^ { 10 } \right) ^ { 3 }$

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

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for a₀, 3a₁, 9a₂, 27a₃, 81a₄, . . . .

determine whether the recurrence relation is a linear homogeneous recurrence relation with constant coeﬃcients. - $a _ { n } - 7 a _ { n - 2 } + a _ { n - 5 } = 0$

solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - $a _ { n } = - 10 a _ { n - 1 } - 21 a _ { n - 2 } , \quad a _ { 0 } = 2 , a _ { 1 } = 1$

Suppose $f ( n ) = 3 f ( n / 2 ) + 1 , f ( 1 ) = 1$ . Find $f ( 8 )$ .

Suppose $f ( n ) = 2 f ( n / 2 ) + 3 , f ( 16 ) = 51$ . Find $\text { Find } f ( 2$ .

Suppose $f ( n ) = 2 f ( n / 2 ) , f ( 8 ) = 2$ . Find $\text { Find } f ( 1 )$ .