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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 $S(m, n)$ satisfies the recurrence relation $S(m, n) = n^{m} − \sum_{k=1 }^{n−1} C(n, k)S(m, k)$ whenever $m \geq n$ and $n > 1,$ with the initial condition $S(m, 1) = 1.$
Let $S(m, n)$ denote the number of onto functions from a set with m elements to a set with $n$ elements. Show that $S(m, n)$ satisfies the recurrence relation$$S(m, n) = ...
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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 $f_{n} = 5f_{n−4} + 3f_{n−5} \:\text{for}\: n = 5, 6, 7,\dots,$ together with the initial conditions $f_{0} = 0, f_{1} = 1, f_{2} = 1, f_{3} = 2, \:\text{and}\: f4 = 3.$ Use this recurrence relation to show that $f_{5n}$ is divisible by $5$, for $n = 1, 2, 3,\dots .$
Show that the Fibonacci numbers satisfy the recurrence relation $f_{n} = 5f_{n−4} + 3f_{n−5} \:\text{for}\: n = 5, 6, 7,\dots,$ together with the initial conditions $...
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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 and tiles of the same color are considered indistinguishable. What are the ... $(A)?$ How many ways are there to lay out a path of seven tiles as described in part $(A)?$
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 and ti...
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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 coverings where the position in the top right corner of the checkerboard ... How many ways are there to completely cover a $2 \times 17$ checkerboard with $1 \times 2$ dominoes?
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 coverings whe...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 25 (Page No. 511)
How many bit sequences of length seven contain an even number of $0s?$
How many bit sequences of length seven contain an even number of $0s?$
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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 $0s.$
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 three of the planes meet in one point, but no four of the planes go through the same point. Find $S_{n}$ using iteration.
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 three of th...
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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 circles (which are the intersections of the sphere and planes passing through ... sphere), if no three of the great circles go through the same point. b) Find $R_{n}$ using iteration.
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 circles (which are...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 21 (Page No. 511)
Find the recurrence relation satisfied by $R_{n},$ where $R_{n}$ is the number of regions that a plane is divided into by $n$ lines, if no two of the lines are parallel and no three of the lines go through the same point. Find $R_{n}$ using iteration.
Find the recurrence relation satisfied by $R_{n},$ where $R_{n}$ is the number of regions that a plane is divided into by $n$ lines, if no two of the lines are parallel a...
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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 of different ways the bus driver can pay a toll of $n$ cents ... which the coins are used matters). In how many different ways can the driver pay a toll of $45$ cents?
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 of ...
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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 signal requires $2$ microseconds. Find a recurrence relation ... initial conditions? How many different messages can be sent in $10$ microseconds using these two signals?
Messages are transmitted over a communications channel using two signals. The transmittal of one signal requires $1$ microsecond, and the transmittal of the other signal ...
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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 conditions? How many ternary strings of length six contain consecutive symbols that are the same?
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 conditions?How many...
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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 conditions? How many ternary strings of length six do not contain consecutive symbols that are the same?
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 conditions?How m...
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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 initial conditions? How many ternary strings of length six contain two consecutive $0s$ or two consecutive $1s?$
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 initial condition...
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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 conditions? How many ternary strings of length six do not contain two consecutive $0s$ or two consecutive $1s?$
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 conditions...
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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 $0s?$
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 le...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 13 (Page No. 511)
A string that contains only $0s, 1s,$ and $2s$ is called a ternary string. Find a recurrence relation for the number of ternary strings of length $n$ that do not contain two consecutive $0s.$ What are the initial conditions? How many ternary strings of length six do not contain two consecutive $0s?$
A string that contains only $0s, 1s,$ and $2s$ is called a ternary string.Find a recurrence relation for the number of ternary strings of length $n$ that do not contain t...
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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. What are the initial conditions? In many ways can this person climb a flight of eight stairs?
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.What are the initial ...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 11 (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 stair or two stairs at a time. What are the initial conditions? In how many ways can this person climb a flight of eight stairs?
Find a recurrence relation for the number of ways to climb n stairs if the person climbing the stairs can take one stair or two stairs at a time.What are the initial cond...
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Kenneth Rosen Edition 7 Exercise 8.1 Question 10 (Page No. 511)
Find a recurrence relation for the number of bit strings of length $n$ that contain the string $01$. What are the initial conditions? How many bit strings of length seven contain the string $01?$
Find a recurrence relation for the number of bit strings of length $n$ that contain the string $01$.What are the initial conditions?How many bit strings of length seven c...
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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 $0s?$
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 l...
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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 $0s?$
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...
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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 $0s?$
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 len...
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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 integer. That is, sequences $a_{1}, a_{2},\dots,a_{k},$ ... How many sequences of the type described in $(A)$ are there when $n$ is an integer with $n \geq 2?$
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 pos...
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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?
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, $20$ pesos, $50$ pesos, and $100$ pesos. Find a recurrence relation for the number of ways to pay a bill of $n$ pesos if the order in which the coins and bills are paid matters.
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, $20$ pesos, $50$ pesos, and...
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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 deposit $n$ dollars in the vending machine, where the order in which ... $10$ for a book of stamps?
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 deposit $n$ ...
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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
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 $n$ ele...
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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.
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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Kenneth Rosen Edition 7 Exercise 6.6 Question 16 (Page No. 439)
The remaining exercises in this section develop another algorithm for generating the permutations of $\{1, 2, 3,\dots,n\}.$ This algorithm is based on Cantor expansions of integers. Every nonnegative integer less than $n!$ has ... between Cantor expansions and permutations as described in the preamble to question $14.$ $3$ $89$ $111$
The remaining exercises in this section develop another algorithm for generating the permutations of $\{1, 2, 3,\dots,n\}.$ This algorithm is based on Cantor expansions o...
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