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Recent questions tagged discrete-mathematics
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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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332
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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May 1, 2020
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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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Kenneth Rosen Edition 7 Exercise 6.6 Question 17 (Page No. 438)
The remaining exercises in this section develop another algorithm for generating the permutations of $\{1, 2, 3,\dots,n\}.$ ... all permutations of a set of n elements based on the correspondence described in the preamble to question $14.$
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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Kenneth Rosen Edition 7 Exercise 6.6 Question 15 (Page No. 438)
Show that the correspondence described in the preamble is a bijection between the set of permutations of $\{1, 2, 3,\dots,n\}$ and the nonnegative integers less than $n!.$
Show that the correspondence described in the preamble is a bijection between the set of permutations of $\{1, 2, 3,\dots,n\}$ and the nonnegative integers less than $n!....
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Kenneth Rosen Edition 7 Exercise 6.6 Question 14 (Page No. 438)
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 a unique ... $a_{1}, a_{2},\dots,a_{n−1}$ that correspond to these permutations. $246531$ $12345$ $654321$
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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Kenneth Rosen Edition 7 Exercise 6.6 Question 13 (Page No. 438)
List all $3$-permutations of $\{1, 2, 3, 4, 5\}.$
List all $3$-permutations of $\{1, 2, 3, 4, 5\}.$
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Kenneth Rosen Edition 7 Exercise 6.6 Question 12 (Page No. 438)
Develop an algorithm for generating the $r$-permutations of a set of $n$ elements.
Develop an algorithm for generating the $r$-permutations of a set of $n$ elements.
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Kenneth Rosen Edition 7 Exercise 6.6 Question 11 (Page No. 438)
Show that Algorithm $3$ produces the next larger $r$-combination in lexicographic order after a given $r$-combination.
Show that Algorithm $3$ produces the next larger $r$-combination in lexicographic order after a given $r$-combination.
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Kenneth Rosen Edition 7 Exercise 6.6 Question 10 (Page No. 438)
Show that Algorithm $1$ produces the next larger permutation in lexicographic order.
Show that Algorithm $1$ produces the next larger permutation in lexicographic order.
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Kenneth Rosen Edition 7 Exercise 6.6 Question 9 (Page No. 438)
Use Algorithm $3$ to list all the $3$-combinations of $\{1, 2, 3, 4, 5\}.$
Use Algorithm $3$ to list all the $3$-combinations of $\{1, 2, 3, 4, 5\}.$
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Kenneth Rosen Edition 7 Exercise 6.6 Question 8 (Page No. 438)
Use Algorithm $2$ to list all the subsets of the set $\{1, 2, 3, 4\}.$
Use Algorithm $2$ to list all the subsets of the set $\{1, 2, 3, 4\}.$
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Kenneth Rosen Edition 7 Exercise 6.6 Question 7 (Page No. 438)
Use Algorithm $1$ to generate the $24$ permutations of the first four positive integers in lexicographic order.
Use Algorithm $1$ to generate the $24$ permutations of the first four positive integers in lexicographic order.
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