Exam 8: A: Advanced Counting Techniques

find a closed form for the generating function for the sequence. -4, 8, 16, 32, 64, . . .

Find the solution of the recurrence relation a_n = 3a_n−1 with a₀ = 2.

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.

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

determine whether the recurrence relation is a linear homogeneous recurrence relation with constant coefficients. - $a _ { n } + a _ { n - 1 } = 1$

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

write the first seven terms of the sequence determined by the generating function. - $( x + 3 ) ^ { 2 }$

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

determine whether the recurrence relation is a linear homogeneous recurrence relation with constant coefficients. - $a _ { n } = 5 a _ { n - 1 } ^ { 2 } - 3 a _ { n - 2 } ^ { 2 }$

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

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

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

find the coefficient of x8 in the power series of each of the function. - $\left( 1 + x ^ { 2 } + x ^ { 4 } + x ^ { 6 } + x ^ { 8 } \right) ^ { 3 }$

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 if each envelope has at least two coins in it.

A market sells ten kinds of soda. You want to buy 12 bottles. How many possibilities are there? if you want at most three bottles of any kind?

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

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

The solutions to $a _ { n } = - 3 a _ { n - 1 } + 18 a _ { n - 2 }$ have the form $a _ { n } = c \cdot 3 ^ { n } + d \cdot ( - 6 ) ^ { n }$ Which of the following are solutions to the given recurrence relation? (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) $a _ { n } = 3 ^ { n } + 6 ^ { n }$

Suppose you use the principle of inclusion-exclusion to find the size of the union of four sets. How many terms must be added or subtracted?