Question 1

Suppose you use the principle of inclusion-exclusion to find the size of the union of four sets. How many terms must be added or subtracted?

Accepted Answer

15.

Question 2

Suppose $| A | = 8 \text { and } | B | = 4$. Find the number of functions $f : A \rightarrow B$ that are onto $B$ .

Accepted Answer

$$4 ^ { 8 } - \left( \begin{array} { l } 
4 \
1
\end{array} \right) 3 ^ { 8 } + \left( \begin{array} { l } 
4 \
2
\end{array} \right) 2 ^ { 8 } - \left( \begin{array} { l } 
4 \
3
\end{array} \right) 1 ^ { 8 } .$$

Question 3

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

Accepted Answer

Yes.

Question 4

Assume that the characteristic equation for a homogeneous linear recurrence relation with constant coeﬃcients
is $$( r + 2 ) ( r + 4 ) ^ { 2 } = 0 \text {. Describe the form for the general solution to the recurrence relation. }$$

Accepted Answer

The answer of Assume that the characteristic equation for a...

Question 5

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

Accepted Answer

The answer of ﬁnd the coeﬃcient of \[x ^ {...

Question 6

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 $$a _ { 0 } = 1$$

Accepted Answer

The answer of Consider the recurrence relation \[a _ {...

Question 7

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

Accepted Answer

The answer of Suppose \(f ( n ) = 2...

Question 8

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 mant elements are in $A \cap B \cap C$

Accepted Answer

The answer of Suppose \(| A | = | B...

Question 9

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 most six coins in it.

Accepted Answer

The answer of Set up a generating function and use...

Question 10

Find the number of positive integers $\leq$ 1000  that are multiples of at least one of  3,5,11 .

Accepted Answer

The answer of Find the number of positive integers $\leq$...

Question 11

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

Accepted Answer

The answer of Find the number of ways to put...

Question 12

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?

Accepted Answer

The answer of A market sells ten kinds of soda....

Question 13

You have ten cards, numbered 1 through 10. In how many ways can you put the ten cards in a row so that
card i is not in spot i, for i = 1, 2, . . . , 10?

Accepted Answer

The answer of You have ten cards, numbered 1 through...

Question 14

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

Accepted Answer

The answer of In questions solve the recurrence relation either...

Question 15

In questions write the ﬁrst seven terms of the sequence determined by the generating function.
-$$( x + 3 ) ^ { 2 } .$$

Accepted Answer

(a) 9, 6,

Question 16

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 $a _ { 0 } , 0 , a _ { 1 } , 0 , a _ { 2 } , 0 , a _ { 3 } , 0 , a _ { 4 } , \ldots$

Accepted Answer

The answer of If  $G ( x )$ ...

Question 17

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 at most three blocks.

Accepted Answer

The answer of Set up a generating function and use...

Question 18

In questions write the ﬁrst seven terms of the sequence determined by the generating function.
-$$( 1 + x ) / ( 1 - x )$$

Accepted Answer

1, 2, 2, 2

Question 19

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.

Accepted Answer

The answer of Consider the recurrence relation \[a _ {...

Question 20

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 that contain a pair of consecutive 0's.

Accepted Answer

The answer of In questions , describe each sequence recursively....

Suppose you use the principle of inclusion-exclusion to find the size of the union of four sets. How many terms must be added or subtracted?

Suppose $| A | = 8 \text { and } | B | = 4$ . Find the number of functions $f : A \rightarrow B$ that are onto $B$ .

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

Assume that the characteristic equation for a homogeneous linear recurrence relation with constant coeﬃcients is $( r + 2 ) ( r + 4 ) ^ { 2 } = 0 \text {. Describe the form for the general solution to the recurrence relation. }$

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

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

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 mant elements are in $A \cap B \cap C$

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 most six coins in it.

Find the number of positive integers $\leq$ 1000 that are multiples of at least one of 3,5,11 .

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

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?

You have ten cards, numbered 1 through 10. In how many ways can you put the ten cards in a row so that card i is not in spot i, for i = 1, 2, . . . , 10?

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

In questions write the ﬁrst seven terms of the sequence determined by the generating function. - $( x + 3 ) ^ { 2 } .$

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 $a _ { 0 } , 0 , a _ { 1 } , 0 , a _ { 2 } , 0 , a _ { 3 } , 0 , a _ { 4 } , \ldots$

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 at most three blocks.

In questions write the ﬁrst seven terms of the sequence determined by the generating function. - $( 1 + x ) / ( 1 - x )$

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 that contain a pair of consecutive 0's.

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

Suppose you use the principle of inclusion-exclusion to find the size of the union of four sets. How many terms must be added or subtracted?

Suppose ∣A∣=8 and ∣B∣=4| A | = 8 \text { and } | B | = 4∣A∣=8 and ∣B∣=4 . Find the number of functions f:A→Bf : A \rightarrow Bf:A→B that are onto BBB .

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

ﬁnd the coeﬃcient of x8x ^ { 8 }x8 in the power series of each of the function. - (1+x2+x4+x6)3\left( 1 + x ^ { 2 } + x ^ { 4 } + x ^ { 6 } \right) ^ { 3 }(1+x2+x4+x6)3

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 f(2)f ( 2 )f(2) .

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 most six coins in it.

Find the number of positive integers ≤\leq≤ 1000 that are multiples of at least one of 3,5,11 .

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

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?

You have ten cards, numbered 1 through 10. In how many ways can you put the ten cards in a row so that card i is not in spot i, for i = 1, 2, . . . , 10?

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

In questions write the ﬁrst seven terms of the sequence determined by the generating function. - (x+3)2.( x + 3 ) ^ { 2 } .(x+3)2.

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 at most three blocks.

In questions write the ﬁrst seven terms of the sequence determined by the generating function. - (1+x)/(1−x)( 1 + x ) / ( 1 - x )(1+x)/(1−x)

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 that contain a pair of consecutive 0's.

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

Suppose $| A | = 8 \text { and } | B | = 4$ . Find the number of functions $f : A \rightarrow B$ that are onto $B$ .

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

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

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

Find the number of positive integers $\leq$ 1000 that are multiples of at least one of 3,5,11 .

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

In questions write the ﬁrst seven terms of the sequence determined by the generating function. - $( x + 3 ) ^ { 2 } .$

In questions write the ﬁrst seven terms of the sequence determined by the generating function. - $( 1 + x ) / ( 1 - x )$

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 that contain a pair of consecutive 0's.