Exam 8: Advanced Counting Techniques

Suppose you have 100 identical marbles and five jars (labeled A, B, C, D, E). In how many ways can you put the marbles in the jars if: (a) each jar has at least six marbles in it? (b) each jar has at most forty marbles in it?

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

How many ways are there to choose five donuts if there are eight varieties and only the type of each donut matters?

In questions , describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. - $a _ { n }$ = the number of ways to go down an n-step staircase if you go down 1, 2, or 3 steps at a time.

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 the oldest child gets three blocks.

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

$\text { If } G ( x ) \text { is the generating function for } a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots \text {, describe in terms of } G ( x ) \text { the generating function for }$ $a _ { 0 } , 3 a _ { 1 } , 9 a _ { 2 } , 27 a _ { 3 } , 81 a _ { 4 } , \ldots$

In questions find a closed form for the generating function for the sequence. -2, 4, 6, 8, 10, 12, . . . .

The Catalan numbers $C _ { n }$ count the number of strings of n +’s and n −’s with the following property: as each string is read from left to right, the number of +’s encountered is always at least as large as the number of −’s . (a) Verify this by listing these strings of lengths 2,4 , and 6 and showing that there are C1, C2, and C3 of these, respectively. (b) Explain how counting these strings is the same as counting the number of ways to correctly parenthesize strings of variables.

In questions write the first seven terms of the sequence determined by the generating function. - $\frac { 1 } { 1 - x } - x ^ { 2 } - x ^ { 3 }$

In questions write the first seven terms of the sequence determined by the generating function. - $x ^ { 2 } / ( 1 - x )$

In questions find a closed form for the generating function for the sequence. - $1,0 , - 1,0,1,0 , - 1,0,1,0 , - 1 , \ldots$

In questions find a closed form for the generating function for the sequence. - $1 , - 1,1 , - 1,1 , - 1,1 , - 1 , \ldots$

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 $F ( n ) = 2 ^ { n }$ ?

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) envelopes A and B have the same number of coins in them.

find the coefficient of $x ^ { 8 }$ in the power series of each of the function. - $\left( 1 + x ^ { 2 } + x ^ { 4 } + x ^ { 6 } + x ^ { 8 } + x ^ { 10 } \right) ^ { 3 }$

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

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 but no more than five coins in it.

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 most five blocks.

Suppose $f ( n ) = 3 f ( n / 2 ) + 1 , f ( 1 ) = 1$ Find $f ( 8 )$ .