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Hot questions in Discrete Mathematics
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1681
Kenneth Rosen Edition 7 Exercise 1.7 Question 17 (Page No. 91)
Show that if $n$ is an integer and $n^3+5$ is odd, then $n$ is even using. a proof by contraposition. a proof by contradiction.
Show that if $n$ is an integer and $n^3+5$ is odd, then $n$ is even using.a proof by contraposition.a proof by contradiction.
Pooja Khatri
2.2k
views
Pooja Khatri
asked
Apr 4, 2019
Mathematical Logic
kenneth-rosen
discrete-mathematics
mathematical-logic
propositional-logic
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0
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1
answer
1682
Kenneth Rosen Edition 7 Exercise 6.1 Question 49 (Page No. 398)
How many bit strings of length $10$ either begin with three $0s$ or end with two $0s?$
How many bit strings of length $10$ either begin with three $0s$ or end with two $0s?$
admin
680
views
admin
asked
Apr 28, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
descriptive
+
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0
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0
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1683
Kenneth Rosen Edition 7 Exercise 6.5 Question 47 (Page No. 433)
Use the product rule to prove Theorem $4,$ by first placing objects in the first box, then placing objects in the second box, and so on.
Use the product rule to prove Theorem $4,$ by first placing objects in the first box, then placing objects in the second box, and so on.
admin
212
views
admin
asked
May 1, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
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0
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0
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1684
Kenneth Rosen Edition 7 Exercise 6.2 Question 25 (Page No. 406)
Describe an algorithm in pseudocode for producing the largest increasing or decreasing subsequence of a sequence of distinct integers.
Describe an algorithm in pseudocode for producing the largest increasing or decreasing subsequence of a sequence of distinct integers.
admin
386
views
admin
asked
Apr 29, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
pigeonhole-principle
descriptive
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0
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0
answers
1685
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...
admin
149
views
admin
asked
May 2, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
descriptive
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0
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1686
Kenneth Rosen Edition 7 Exercise 6.4 Question 34 (Page No. 422)
Use question $33$ to give an alternative proof of Corollary $2$ in Section $6.3,$ which states that $\binom{n}{k} = \binom{n}{n−k} $ whenever $k$ is an integer with $0 \leq k \leq n.[$Hint: Consider the number of paths of the type described in question $33$ ... $(0, 0)\: \text{to}\:(k, n − k).]$
Use question $33$ to give an alternative proof of Corollary $2$ in Section $6.3,$ which states that $\binom{n}{k} = \binom{n}{n−k} $ whenever $k$ is an integer with $0 ...
admin
231
views
admin
asked
Apr 30, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
descriptive
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0
votes
0
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1687
Kenneth Rosen Edition 7 Exercise 6.2 Question 46 (Page No. 407)
Let $n_{1}, n_{2},\dots,n_{t}$ be positive integers. Show that if $n_{1} + n_{2} +\dots + n_{t} − t + 1$ objects are placed into $t$ boxes, then for some $i, i = 1, 2,\dots,t,$ the $i^{\text{th}}$ box contains at least $n_{i}$ objects.
Let $n_{1}, n_{2},\dots,n_{t}$ be positive integers. Show that if $n_{1} + n_{2} +\dots + n_{t} − t + 1$ objects are placed into $t$ boxes, then for some $i, i = 1, 2,\...
admin
265
views
admin
asked
Apr 29, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
pigeonhole-principle
descriptive
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0
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0
answers
1688
Kenneth Rosen Edition 7 Exercise 6.5 Question 55 (Page No. 434)
How many ways are there to distribute six indistinguishable objects into four indistinguishable boxes so that each of the boxes contains at least one object?
How many ways are there to distribute six indistinguishable objects into four indistinguishable boxes so that each of the boxes contains at least one object?
admin
227
views
admin
asked
May 1, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
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0
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0
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1689
Kenneth Rosen Edition 7 Exercise 6.1 Question 43 (Page No. 397)
How many $4$-element $RNA$ sequences contain the base $U?$ do not contain the sequence $CUG?$ do not contain all four bases $A, U, C,$ and $G?$ contain exactly two of the four bases $A, U, C,$ and $G?$
How many $4$-element $RNA$ sequencescontain the base $U?$ do not contain the sequence $CUG?$ do not contain all four bases $A, U, C,$ and $G?$ contain exactly two of the ...
admin
412
views
admin
asked
Apr 28, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
descriptive
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1690
Kenneth Rosen Edition 7 Exercise 6.1 Question 32 (Page No. 397)
How many strings of eight uppercase English letters are there if letters can be repeated? if no letter can be repeated? that start with $X,$ if letters can be repeated? that start with $X,$ if no letter can be repeated? that ... can be repeated? that start or end with the letters $BO$ (in that order), if letters can be repeated?
How many strings of eight uppercase English letters are thereif letters can be repeated?if no letter can be repeated?that start with $X,$ if letters can be repeated?that ...
admin
549
views
admin
asked
Apr 28, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
descriptive
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0
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0
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1691
Kenneth Rosen Edition 7 Exercise 8.1 Question 38 (Page No. 512)
Question $38-45$ involve the Reve's puzzle, the variation of the Tower of Hanoi puzzle with four pegs and $n$ disks. Before presenting these exercises, we describe the Frame-Stewart algorithm for moving the disks from peg $1$ to ... are moved. Show that the Reve's puzzle with three disks can be solved using five, and no fewer, moves.
Question $38–45$ involve the Reve’s puzzle, the variation of the Tower of Hanoi puzzle with four pegs and $n$ disks. Before presenting these exercises, we describe th...
admin
209
views
admin
asked
May 2, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
descriptive
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0
votes
0
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1692
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...
admin
156
views
admin
asked
May 2, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
descriptive
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0
votes
0
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1693
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...
admin
151
views
admin
asked
May 2, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
descriptive
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1694
Kenneth Rosen Edition 7 Exercise 8.1 Question 51 (Page No. 512)
Let $\{a_{n}\}$ be a sequence of real numbers. The backward differences of this sequence are defined recursively as shown next. The first difference $\triangledown a_{n}$ is $\triangledown a_{n} = a_{n} − a_{n−1}.$ ... $a_{n}, \triangledown a_{n},\: \text{and}\: \triangledown^{2}a_{n}.$
Let $\{a_{n}\}$ be a sequence of real numbers. The backward differences of this sequence are defined recursively as shown next. The first difference $\triangledown a_{n}$...
admin
134
views
admin
asked
May 3, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
descriptive
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0
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0
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1695
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. Use the recurrence relation developed in Example $5$ to calculate $C_{4},$ ... $(B)$ by finding $C_{4},$ using the closed formula for $C_{n}$ mentioned in the solution of Example $5.$
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.Use the recurrenc...
admin
148
views
admin
asked
May 2, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
descriptive
+
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0
votes
0
answers
1696
Kenneth Rosen Edition 7 Exercise 6.5 Question 51 (Page No. 434)
How many ways are there to distribute six distinguishable objects into four indistinguishable boxes so that each of the boxes contains at least one object?
How many ways are there to distribute six distinguishable objects into four indistinguishable boxes so that each of the boxes contains at least one object?
admin
192
views
admin
asked
May 1, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
+
–
0
votes
0
answers
1697
Kenneth Rosen Edition 7 Exercise 8.1 Question 45 (Page No. 512)
Question $38-45$ involve the Reve's puzzle, the variation of the Tower of Hanoi puzzle with four pegs and $n$ disks. Before presenting these exercises, we describe the Frame-Stewart algorithm for moving the disks from peg $1$ to peg $4$ so that no ... how the disks are moved. Show that $R(n)\: \text{is}\: O(\sqrt{n}2^{\sqrt{2n}}).$
Question $38–45$ involve the Reve’s puzzle, the variation of the Tower of Hanoi puzzle with four pegs and $n$ disks. Before presenting these exercises, we describe th...
admin
157
views
admin
asked
May 2, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
descriptive
+
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0
votes
0
answers
1698
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 $\{1, 2, 3, 4, 5\}.$
admin
172
views
admin
asked
May 1, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
+
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0
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0
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1699
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 order of multiplication. Check your result with the closed formula for $C_{5}$ mentioned in the solution of Example $5.$
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 order of m...
admin
137
views
admin
asked
May 2, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
descriptive
+
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0
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0
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1700
Kenneth Rosen Edition 7 Exercise 6.1 Question 60 (Page No. 398)
A key in the Vigenère cryptosystem is a string of English letters, where the case of the letters does not matter. How many different keys for this cryptosystem are there with three, four, five, or six letters?
A key in the Vigenère cryptosystem is a string of English letters, where the case of the letters does not matter. How many different keys for this cryptosystem are there...
admin
394
views
admin
asked
Apr 28, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
descriptive
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