Question 1

How many permutations of all 26 letters of the alphabet are there that contain at least one of the words CART, SHOW, LIKE?

Accepted Answer

The answer of How many permutations of all 26 letters...

Question 2

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

Accepted Answer

The answer of ﬁnd the coeﬃcient of x8 in the...

Question 3

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for a₃, a₄, a₅, a₆, . . . .

Accepted Answer

The answer of If G(x) is the generating function for...

Question 4

How many permutations of all 26 letters of the alphabet are there that contain at least one of the words: CAR, CARE, SCAR, SCARE?

Accepted Answer

The answer of How many permutations of all 26 letters...

Question 5

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 6

An oﬃce manager has four employees and nine reports to be done. In how many ways can the reports be assigned to the employees so that each employee has at least one report to do.

Accepted Answer

The answer of An oﬃce manager has four employees and...

Question 7

How many permutations of the 26 letters of the alphabet are there that do not contain any of the following strings: LOP, SLOP, SLOPE, LOPE.

Accepted Answer

The answer of How many permutations of the 26 letters...

Question 8

ﬁnd a closed form for the generating function for the sequence.
-$$\left( \begin{array} { c } 
50 \
1
\end{array} \right) , 2 \left( \begin{array} { c } 
50 \
2
\end{array} \right) , 3 \left( \begin{array} { c } 
50 \
3
\end{array} \right) , \ldots , 50 \left( \begin{array} { c } 
50 \
50
\end{array} \right) , 0,0,0 , \ldots$$

Accepted Answer

The answer of ﬁnd a closed form for the generating...

Question 9

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

Accepted Answer

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

Question 10

Consider the recurrence relation $a _ { n } = 2 a _ { n - 1 } + 1$ (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 11

Use generating functions to solve $$a _ { n } = 7 a _ { n - 1 } - 10 a _ { n - 2 } , a _ { 0 } = 1 , a _ { 1 } = 1$$

Accepted Answer

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

Question 12

Find the number of ways to climb a 12-step staircase, if you go up either one or three steps at a time.

Accepted Answer

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

Question 13

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

Accepted Answer

The answer of ﬁnd a closed form for the generating...

Question 14

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

Accepted Answer

The answer of solve the recurrence relation either by using...

Question 15

An oﬃce manager has ﬁve employees and 12 projects to be completed. In how many ways can the projects be assigned to the employees so that each employee works on at least one project.

Accepted Answer

The answer of An oﬃce manager has ﬁve employees and...

Question 16

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

Accepted Answer

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

Question 17

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

Accepted Answer

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

Question 18

write the ﬁrst seven terms of the sequence determined by the generating function.
-5

Accepted Answer

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

Question 19

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

Accepted Answer

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

Question 20

solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms.
-$$a _ { n } = 5 a _ { n - 1 } - 4 a _ { n - 2 } , \quad a _ { 0 } = 1 , a _ { 1 } = 0$$

Accepted Answer

The answer of solve the recurrence relation either by using...

Exam 8: A: Advanced Counting Techniques

How many permutations of all 26 letters of the alphabet are there that contain at least one of the words CART, SHOW, LIKE?

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

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for a₃, a₄, a₅, a₆, . . . .

How many permutations of all 26 letters of the alphabet are there that contain at least one of the words: CAR, CARE, SCAR, SCARE?

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

An oﬃce manager has four employees and nine reports to be done. In how many ways can the reports be assigned to the employees so that each employee has at least one report to do.

How many permutations of the 26 letters of the alphabet are there that do not contain any of the following strings: LOP, SLOP, SLOPE, LOPE.

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

Use generating functions to solve $a _ { n } = 7 a _ { n - 1 } - 10 a _ { n - 2 } , a _ { 0 } = 1 , a _ { 1 } = 1$

Find the number of ways to climb a 12-step staircase, if you go up either one or three steps at a time.

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

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

An oﬃce manager has ﬁve employees and 12 projects to be completed. In how many ways can the projects be assigned to the employees so that each employee works on at least one project.

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

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

write the ﬁrst seven terms of the sequence determined by the generating function. -5

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

solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - $a _ { n } = 5 a _ { n - 1 } - 4 a _ { n - 2 } , \quad a _ { 0 } = 1 , a _ { 1 } = 0$

Exam 8: A: Advanced Counting Techniques

How many permutations of all 26 letters of the alphabet are there that contain at least one of the words CART, SHOW, LIKE?

ﬁnd the coeﬃcient of 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

If G(x) is the generating function for a0, a1, a2, a3, . . . , describe in terms of G(x) the generating function for a3, a4, a5, a6, . . . .

How many permutations of all 26 letters of the alphabet are there that contain at least one of the words: CAR, CARE, SCAR, SCARE?

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

An oﬃce manager has four employees and nine reports to be done. In how many ways can the reports be assigned to the employees so that each employee has at least one report to do.

How many permutations of the 26 letters of the alphabet are there that do not contain any of the following strings: LOP, SLOP, SLOPE, LOPE.

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 ?

Use generating functions to solve an=7an−1−10an−2,a0=1,a1=1a _ { n } = 7 a _ { n - 1 } - 10 a _ { n - 2 } , a _ { 0 } = 1 , a _ { 1 } = 1an​=7an−1​−10an−2​,a0​=1,a1​=1

Find the number of ways to climb a 12-step staircase, if you go up either one or three steps at a time.

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

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

An oﬃce manager has ﬁve employees and 12 projects to be completed. In how many ways can the projects be assigned to the employees so that each employee works on at least one project.

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

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

write the ﬁrst seven terms of the sequence determined by the generating function. -5

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

solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - an=5an−1−4an−2,a0=1,a1=0a _ { n } = 5 a _ { n - 1 } - 4 a _ { n - 2 } , \quad a _ { 0 } = 1 , a _ { 1 } = 0an​=5an−1​−4an−2​,a0​=1,a1​=0

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

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for a₃, a₄, a₅, a₆, . . . .

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

Use generating functions to solve $a _ { n } = 7 a _ { n - 1 } - 10 a _ { n - 2 } , a _ { 0 } = 1 , a _ { 1 } = 1$

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

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

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

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

solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - $a _ { n } = 5 a _ { n - 1 } - 4 a _ { n - 2 } , \quad a _ { 0 } = 1 , a _ { 1 } = 0$