Exam 8: Advanced Counting Techniques

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.

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 no envelope is empty.

Find the number of positive integers $\leq$ 1000 that are multiples of at least one of 3,4,12 .

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

In questions find a closed form for the generating function for the sequence. - $1 , \frac { 1 } { 2 ! } , \frac { 1 } { 4 ! } , \frac { 1 } { 6 ! } , \frac { 1 } { 8 ! } , \ldots$

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.

In questions 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, . . . .

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 begin with 1.

find the coefficient of $x ^ { 8 }$ in the power series of each of the function. - $x ^ { 3 } / ( 1 - 3 x )$

In questions write the first seven terms of the sequence determined by the generating function. - $1 / ( 1 - 3 x )$

In questions find a closed form for the generating function for the sequence. -2, 0, 0, 2, 0, 0, 2, 0, 0, 2, . . . .

In questions write the first seven terms of the sequence determined by the generating function. - $( 1 + x ) ^ { 9 }$

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

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

Thesolutionssolutionsto thetogivanen=recurrence−3an−1 +relation?18an−2 have the form $a _ { n } = c \cdot 3 ^ { n } + d \cdot ( - 6 ) ^ { n }$ Which of the following are (a) $a _ { n } = 3 ^ { n + 1 } + ( - 6 ) ^ { n }$ . (b) $a _ { n } = 5 ( - 6 ) ^ { n }$ . (c) $a _ { n } = 3 c - 6 d$ . (d) $a _ { n } = 3 ^ { n - 2 }$ . (e) $a _ { n } = \pi \left( 3 ^ { n } + ( - 6 ) ^ { n } \right)$ . (f) $a _ { n } = - 3 ^ { n }$ . (g) $a _ { n } = 3 ^ { n } \left( 1 + ( - 2 ) ^ { n } \right)$ (h) $a _ { n } = 3 ^ { n } + 6 ^ { n }$ .

How many permutations of all 26 letters of the alphabet are there that contain none of the words: SAVE, PLAY, SNOW?

Find the number of positive integers $\leq$ 1000 that are multiples of at least one of 2,6,12 .

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

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

determine whether the recurrence relation is a linear homogeneous recurrence relation with constant coefficients. - $a _ { n } = 0.7 a _ { n - 1 } - 0.3 a _ { n - 2 } \text {. }$