Exam 8: A: Advanced Counting Techniques

You have ten cards, numbered 1 through 10. In how many ways can you put the ten cards in a row so that card i is not in spot i, for i = 1, 2, . . . , 10?

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

find a closed form for the generating function for the sequence. -1, 12!, 14!, 16!, 18! . . .

A vending machine dispensing books of stamps accepts only $1 coins, $1 bills, and $2 bills. Let an denote

A market sells 40 kinds of candy bars. You want to buy 15 candy bars. (a) How many possibilities are there? (b) How many possibilities are there if you want at least three peanut butter bars and at least five almond bars? (c) How many possibilities are there if you want exactly three peanut butter bars and exactly five almond bars? (d) How many possibilities are there if you want at most four toffee bars and at most six mint bars?

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for a0, 0, 0, a₁, 0, 0, a₂, 0, 0, a₃, . . . .

find a closed form for the generating function for the sequence. -2, 3, 4, 5, 6, 7, . . .

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

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

find the coefficient of x8 in the power series of each of the function. - $x ^ { 3 } / ( 1 - 3 x )$

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

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

Find the number of bit strings of length eight that contain a pair of consecutive 0's.

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 an even number of coins in it.

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

find a closed form for the generating function for the sequence. - $\left( \begin{array} { l } 50 \\50\end{array} \right) , \left( \begin{array} { c } 50 \\49\end{array} \right) , \left( \begin{array} { c } 50 \\48\end{array} \right) , \ldots , \left( \begin{array} { c } 50 \\1\end{array} \right) , \left( \begin{array} { c } 50 \\0\end{array} \right) , 0,0,0 , \ldots$

find the coefficient of x8 in the power series of each of the function. - $1 / ( 1 - x ) ^ { 2 }$

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

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

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 two blocks.