Exam 8: 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?

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 $0,0,0 , a _ { 3 } , a _ { 4 } , a _ { 5 } , \ldots$

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

In questions write the first seven terms of the sequence determined by the generating function. - $( 1 + x ) ^ { 5 }$

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

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 $a _ { 3 } , a _ { 4 } , a _ { 5 } , a _ { 6 } , \ldots$

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.

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 least one block.

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

determine whether the recurrence relation is a linear homogeneous recurrence relation with constant coefficients. - $a _ { n } = a _ { n - 3 }$

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) if envelope A has at least three coins in it.

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

Assume that the characteristic equation for a homogeneous linear recurrence relation with constant coefficients $\text { is } ( r - 3 ) ^ { 2 } ( r - 4 ) ^ { 3 } ( r + 7 ) ^ { 2 } = 0 \text {. Describe the form for the general solution to the recurrence relation. }$

A vending machine dispensing books of stamps accepts only $1 coins, $1 bills, and $2 bills. Let an denote the number of ways of depositing n dollars in the vending machine, where the order in which the coins and bills are deposited matters. (a) Find a recurrence relation for an and give the necessary initial condition(s). (b) Find an explicit formula for an by solving the recurrence relation in part (a).

find the coefficient of $x ^ { 8 }$ in the power series of each of the function. - $\left( 1 + x ^ { 3 } \right) ^ { 12 }$

find the coefficient of $x ^ { 8 }$ in the power series of each of the function. - $( 1 + x ) \left( 1 + x ^ { 2 } \right) \left( 1 + x ^ { 3 } \right) \left( 1 + x ^ { 4 } \right) \left( 1 + x ^ { 5 } \right)$

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

How many permutations of all 26 letters of the alphabet are there that contain at least one of the words SWORD, PLANT, CARTS?

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

find the coefficient of $x ^ { 8 }$ in the power series of each of the function. - $1 / ( 1 - x ) ^ { 2 }$