Question 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?

Accepted Answer

The answer of Suppose you have 100 identical marbles and...

Question 2

Suppose $f ( n ) = f ( n / 3 ) + 2 n , f ( 1 ) = 1$. Find $f ( 27 )$

Accepted Answer

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

Question 3

How many ways are there to choose ﬁve donuts if there are eight varieties and only the type of each donut
matters?

Accepted Answer

The answer of How many ways are there to choose...

Question 4

In questions , describe each sequence recursively. Include initial conditions and assume that the sequences
begin with a1.
-$$a _ { n }$$ = the number of ways to go down an n-step staircase if you go down 1, 2, or 3 steps at a time.

Accepted Answer

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

Question 5

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 three blocks.

Accepted Answer

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

Question 6

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

Accepted Answer

The answer of Use generating functions to solve \(a _...

Question 7

$$\text { If } G ( x ) \text { is the generating function for } a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots \text {, describe in terms of } G ( x ) \text { the generating function for }$$ $$a _ { 0 } , 3 a _ { 1 } , 9 a _ { 2 } , 27 a _ { 3 } , 81 a _ { 4 } , \ldots$$

Accepted Answer

The answer of \[\text { If } G ( x...

Question 8

In questions ﬁnd a closed form for the generating function for the sequence.
-2, 4, 6, 8, 10, 12, . . . .

Accepted Answer

The answer of In questions ﬁnd a closed form for...

Question 9

The Catalan numbers $C _ { n }$ count the number of strings of n +’s  and n −’s  with the following property: as each string is read from left to right, the number of  +’s encountered is always at least as large as the number of  −’s . (a) Verify this by listing these strings of lengths 2,4 , and 6 and showing that there are C1, C2, and C3 of  these, respectively.
(b) Explain how counting these strings is the same as counting the number of ways to correctly parenthesize strings of variables.

Accepted Answer

The answer of The Catalan numbers \(C _ { n...

Question 10

In questions write the ﬁrst seven terms of the sequence determined by the generating function.
-$$\frac { 1 } { 1 - x } - x ^ { 2 } - x ^ { 3 }$$

Accepted Answer

The answer of In questions write the ﬁrst seven terms...

Question 11

In questions write the ﬁrst seven terms of the sequence determined by the generating function.
-$$x ^ { 2 } / ( 1 - x )$$

Accepted Answer

The answer of In questions write the ﬁrst seven terms...

Question 12

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

Accepted Answer

The answer of In questions ﬁnd a closed form for...

Question 13

In questions ﬁnd a closed form for the generating function for the sequence.
-$$1 , - 1,1 , - 1,1 , - 1,1 , - 1 , \ldots$$

Accepted Answer

The answer of In questions ﬁnd a closed form for...

Question 14

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 $F ( n ) = 2 ^ { n }$ ?

Accepted Answer

The answer of What form does a particular solution of...

Question 15

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 (labeled A, B, C) envelopes A and B have the same number of coins in them.

Accepted Answer

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

Question 16

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

Accepted Answer

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

Question 17

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

Accepted Answer

The answer of How many positive integers not exceeding 1000...

Question 18

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.

Accepted Answer

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

Question 19

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.

Accepted Answer

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

Question 20

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

Accepted Answer

The answer of Suppose \(f ( n ) = 3...

Exam 8: Advanced Counting Techniques

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?

Suppose $f ( n ) = f ( n / 3 ) + 2 n , f ( 1 ) = 1$ . Find $f ( 27 )$

How many ways are there to choose ﬁve donuts if there are eight varieties and only the type of each donut matters?

In questions , describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. - $a _ { n }$ = the number of ways to go down an n-step staircase if you go down 1, 2, or 3 steps at a time.

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 three blocks.

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

$\text { If } G ( x ) \text { is the generating function for } a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots \text {, describe in terms of } G ( x ) \text { the generating function for }$ $a _ { 0 } , 3 a _ { 1 } , 9 a _ { 2 } , 27 a _ { 3 } , 81 a _ { 4 } , \ldots$

In questions ﬁnd a closed form for the generating function for the sequence. -2, 4, 6, 8, 10, 12, . . . .

In questions write the ﬁrst seven terms of the sequence determined by the generating function. - $\frac { 1 } { 1 - x } - x ^ { 2 } - x ^ { 3 }$

In questions write the ﬁrst seven terms of the sequence determined by the generating function. - $x ^ { 2 } / ( 1 - x )$

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

In questions ﬁnd a closed form for the generating function for the sequence. - $1 , - 1,1 , - 1,1 , - 1,1 , - 1 , \ldots$

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 $F ( n ) = 2 ^ { n }$ ?

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 (labeled A, B, C) envelopes A and B have the same number of coins in them.

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

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

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.

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

Exam 8: Advanced Counting Techniques

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?

Suppose f(n)=f(n/3)+2n,f(1)=1f ( n ) = f ( n / 3 ) + 2 n , f ( 1 ) = 1f(n)=f(n/3)+2n,f(1)=1 . Find f(27)f ( 27 )f(27)

How many ways are there to choose ﬁve donuts if there are eight varieties and only the type of each donut matters?

In questions , describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. - ana _ { n }an​ = the number of ways to go down an n-step staircase if you go down 1, 2, or 3 steps at a time.

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 three blocks.

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

In questions ﬁnd a closed form for the generating function for the sequence. -2, 4, 6, 8, 10, 12, . . . .

In questions write the ﬁrst seven terms of the sequence determined by the generating function. - 11−x−x2−x3\frac { 1 } { 1 - x } - x ^ { 2 } - x ^ { 3 }1−x1​−x2−x3

In questions write the ﬁrst seven terms of the sequence determined by the generating function. - x2/(1−x)x ^ { 2 } / ( 1 - x )x2/(1−x)

In questions ﬁnd a closed form for the generating function for the sequence. - 1,0,−1,0,1,0,−1,0,1,0,−1,…1,0 , - 1,0,1,0 , - 1,0,1,0 , - 1 , \ldots1,0,−1,0,1,0,−1,0,1,0,−1,…

In questions ﬁnd a closed form for the generating function for the sequence. - 1,−1,1,−1,1,−1,1,−1,…1 , - 1,1 , - 1,1 , - 1,1 , - 1 , \ldots1,−1,1,−1,1,−1,1,−1,…

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)=2nF ( n ) = 2 ^ { n }F(n)=2n ?

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 (labeled A, B, C) envelopes A and B have the same number of coins in them.

ﬁnd the coeﬃcient of x8x ^ { 8 }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

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

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.

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 $f ( n ) = f ( n / 3 ) + 2 n , f ( 1 ) = 1$ . Find $f ( 27 )$

In questions , describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. - $a _ { n }$ = the number of ways to go down an n-step staircase if you go down 1, 2, or 3 steps at a time.

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

In questions write the ﬁrst seven terms of the sequence determined by the generating function. - $\frac { 1 } { 1 - x } - x ^ { 2 } - x ^ { 3 }$

In questions write the ﬁrst seven terms of the sequence determined by the generating function. - $x ^ { 2 } / ( 1 - x )$

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

In questions ﬁnd a closed form for the generating function for the sequence. - $1 , - 1,1 , - 1,1 , - 1,1 , - 1 , \ldots$

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 $F ( n ) = 2 ^ { n }$ ?

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

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