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Kenneth Rosen Edition 7 Exercise 8.1 Question 35 (Page No. 512)
Question $33-37$ deal with a variation of the $\textbf{Josephus problem}$ described by Graham, Knuth, and Patashnik in $[G_{r}K_{n}P_{a}94].$ This problem is based on an account by ...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 34 (Page No. 512)
Question $33-37$ deal with a variation of the $\textbf{Josephus problem}$ described by Graham, Knuth, and Patashnik in $[G_{r}K_{n}P_{a}94].$ This problem is ... $m$ is a nonnegative integer and $k$ is a nonnegative integer less than $2m.]$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 33 (Page No. 512)
Question $33-37$ deal with a variation of the $\textbf{Josephus problem}$ described by Graham, Knuth, and Patashnik in $[G_{r}K_{n}P_{a}94].$ This problem is based on an ... value of $J (n)$ for each integer $n$ with $1 \leq n \leq 16.$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 32 (Page No. 512)
In the Tower of Hanoi puzzle, suppose our goal is to transfer all $n$ disks from peg $1$ to peg $3,$ but we cannot move a disk directly between ... allowable arrangement of the n disks occurs in the solution of this variation of the puzzle.
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Kenneth Rosen Edition 7 Exercise 8.1 Question 31 (Page No. 512)
Use the recurrence relation developed in Example $5$ to determine $C_{5},$ the number of ways to parenthesize the product of six numbers so as to determine the ... the closed formula for $C_{5}$ mentioned in the solution of Example $5.$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 30 (Page No. 512)
Write out all the ways the product $x_{0} \cdot x_{1} \cdot x_{2} \cdot x_{3} \cdot x_{4}$ can be parenthesized to determine the order of multiplication. ... $C_{n}$ mentioned in the solution of Example $5.$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 29 (Page No. 512)
Let $S(m, n)$ denote the number of onto functions from a set with m elements to a set with $n$ elements. Show that $S(m, n)$ ... $n > 1,$ with the initial condition $S(m, 1) = 1.$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 28 (Page No. 512)
Show that the Fibonacci numbers satisfy the recurrence relation $f_{n} = 5f_{n−4} + 3f_{n−5} \:\text{for}\: n = 5, 6, 7,\dots,$ ... that $f_{5n}$ is divisible by $5$, for $n = 1, 2, 3,\dots .$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 27 (Page No. 512)
Find a recurrence relation for the number of ways to lay out a walkway with slate tiles if the tiles are red, green, or gray, so that no two red tiles are adjacent ... are there to lay out a path of seven tiles as described in part $(A)?$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 26 (Page No. 512)
Find a recurrence relation for the number of ways to completely cover a $2 \times n$ checkerboard with $1 \times 2$ dominoes. [Hint: Consider separately the ... completely cover a $2 \times 17$ checkerboard with $1 \times 2$ dominoes?
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Kenneth Rosen Edition 7 Exercise 8.1 Question 24 (Page No. 511)
Find a recurrence relation for the number of bit sequences of length $n$ with an even number of $0s.$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 23 (Page No. 511)
Find the recurrence relation satisfied by $S_{n},$ where $S_{n}$ is the number of regions into which three-dimensional space is divided by $n$ planes if every ... four of the planes go through the same point.Find $S_{n}$ using iteration.
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Kenneth Rosen Edition 7 Exercise 8.1 Question 22 (Page No. 511)
a) Find the recurrence relation satisfied by $R_{n},$ where $R_{n}$ is the number of regions into which the surface of a sphere is divided by $n$ great ... of the great circles go through the same point.b) Find $R_{n}$ using iteration.
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Kenneth Rosen Edition 7 Exercise 8.1 Question 20 (Page No. 511)
A bus driver pays all tolls, using only nickels and dimes, by throwing one coin at a time into the mechanical toll collector.Find a recurrence relation for the number ... .In how many different ways can the driver pay a toll of $45$ cents?
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Kenneth Rosen Edition 7 Exercise 8.1 Question 19 (Page No. 511)
Messages are transmitted over a communications channel using two signals. The transmittal of one signal requires $1$ microsecond, and the transmittal of the other ... messages can be sent in $10$ microseconds using these two signals?
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Kenneth Rosen Edition 7 Exercise 8.1 Question 18 (Page No. 511)
Find a recurrence relation for the number of ternary strings of length $n$ that contain two consecutive symbols that are the same.What are the initial ... many ternary strings of length six contain consecutive symbols that are the same?
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Kenneth Rosen Edition 7 Exercise 8.1 Question 17 (Page No. 511)
Find a recurrence relation for the number of ternary strings of length $n$ that do not contain consecutive symbols that are the same.What are the initial ... strings of length six do not contain consecutive symbols that are the same?
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Kenneth Rosen Edition 7 Exercise 8.1 Question 16 (Page No. 511)
Find a recurrence relation for the number of ternary strings of length $n$ that contain either two consecutive $0s$ or two consecutive $1s.$What are the ... strings of length six contain two consecutive $0s$ or two consecutive $1s?$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 15 (Page No. 511)
Find a recurrence relation for the number of ternary strings of length n that do not contain two consecutive $0s$ or two consecutive $1s.$ What are the initial ... of length six do not contain two consecutive $0s$ or two consecutive $1s?$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 14 (Page No. 511)
Find a recurrence relation for the number of ternary strings of length n that contain two consecutive $0s.$ What are the initial conditions?How many ternary strings of length six contain two consecutive $0s?$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 12 (Page No. 511)
Find a recurrence relation for the number of ways to climb $n$ stairs if the person climbing the stairs can take one, two, or three stairs at a time. ... the initial conditions?In many ways can this person climb a flight of eight stairs?
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Kenneth Rosen Edition 7 Exercise 8.1 Question 9 (Page No. 511)
Find a recurrence relation for the number of bit strings of length n that do not contain three consecutive $0s.$ What are the initial conditions?How many bit strings of length seven do not contain three consecutive $0s?$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 8 (Page No. 511)
Find a recurrence relation for the number of bit strings of length $n$ that contain three consecutive $0s.$ What are the initial conditions?How many bit strings of length seven contain three consecutive $0s?$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 7 (Page No. 510 - 511)
Find a recurrence relation for the number of bit strings of length $n$ that contain a pair of consecutive $0s$.What are the initial conditions?How many bit strings of length seven contain two consecutive $0s?$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 6 (Page No. 510)
Find a recurrence relation for the number of strictly increasing sequences of positive integers that have 1 as their first term and n as their last term, where n is a positive ... $(A)$ are there when $n$ is an integer with $n \geq 2?$
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Kenneth Rosen Edition 7 Exercise 8.1 Question 5 (Page No. 510)
How many ways are there to pay a bill of $17$ pesos using the currency described in question $4,$ where the order in which coins and bills are paid matters?
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Kenneth Rosen Edition 7 Exercise 8.1 Question 4 (Page No. 510)
A country uses as currency coins with values of $1$ peso, $2$ pesos, $5$ pesos, and $10$ pesos and bills with values of $5$ pesos, $10$ pesos, $20$ ... pay a bill of $n$ pesos if the order in which the coins and bills are paid matters.
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Kenneth Rosen Edition 7 Exercise 8.1 Question 3 (Page No. 510)
A vending machine dispensing books of stamps accepts only one-dollar coins, $\$1$ bills, and $\$5$ bills.Find a recurrence relation for the number of ways to ... $10$ for a book of stamps?
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Kenneth Rosen Edition 7 Exercise 8.1 Question 2 (Page No. 510)
Find a recurrence relation for the number of permutations of a set with $n$ elements.Use this recurrence relation to find the number of permutations of a set with $n$ elements using iteration
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Kenneth Rosen Edition 7 Exercise 8.1 Question 1 (Page No. 510)
Use mathematical induction to verify the formula derived in Example $2$ for the number of moves required to complete the Tower of Hanoi puzzle.
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