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.

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

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

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.

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. - $a _ { n } = 5 a _ { n - 1 } - 4 a _ { n - 2 } , \quad a _ { 0 } = 1 , a _ { 1 } = 0$