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Hot questions in Discrete Mathematics
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2011
Kenneth Rosen Edition 7 Exercise 8.1 Question 46 (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 $\triangledown a_{n} = a_{n} − a_{n−1}.$ The $(k + 1)^{\text{st}}$ ... $a_{n} = 4.$ $a_{n} = 2n.$ $a_{n} = n^{2}.$ $a_{n} = 2^{n}.$
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
146
views
admin
asked
May 3, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
descriptive
+
–
0
votes
0
answers
2012
Kenneth Rosen Edition 7 Exercise 6.2 Question 42 (Page No. 406)
Is the statement in question $41$ true if $24$ is replaced by $2?$ $23?$ $25?$ $30?$
Is the statement in question $41$ true if $24$ is replaced by$2?$$23?$$25?$$30?$
admin
281
views
admin
asked
Apr 29, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
pigeonhole-principle
descriptive
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0
votes
0
answers
2013
Kenneth Rosen Edition 7 Exercise 6.5 Question 50 (Page No. 434)
How many ways are there to distribute five distinguishable objects into three indistinguishable boxes?
How many ways are there to distribute five distinguishable objects into three indistinguishable boxes?
admin
204
views
admin
asked
May 1, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
+
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0
votes
0
answers
2014
Kenneth Rosen Edition 7 Exercise 6.2 Question 34 (Page No. 406)
Assuming that no one has more than $1,000,000$ hairs on the head of any person and that the population of New York City was $8,008,278\:\text{in}\: 2010,$ show there had to be at least nine people in NewYork City in $2010$ with the same number of hairs on their heads.
Assuming that no one has more than $1,000,000$ hairs on the head of any person and that the population of New York City was $8,008,278\:\text{in}\: 2010,$ show there had ...
admin
285
views
admin
asked
Apr 29, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
pigeonhole-principle
descriptive
+
–
0
votes
0
answers
2015
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 $0s.$
admin
189
views
admin
asked
May 2, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
descriptive
+
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0
votes
0
answers
2016
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...
admin
223
views
admin
asked
May 2, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
descriptive
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0
votes
0
answers
2017
Kenneth Rosen Edition 7 Exercise 6.5 Question 28 (Page No. 433)
Show that there are $C(n + r − q_{1} − q_{2} −\dots − q_{r} −1, n − q_{1} − q_{2} −\dots − q_{r})$ different unordered selections of $n$ objects of $r$ different types that include at least $q_{1}$ objects of type one, $q_{2}$ objects of type two $,\dots, $ and $q_{r}$ objects of type $r.$
Show that there are $C(n + r − q_{1} − q_{2} −\dots − q_{r} −1, n − q_{1} − q_{2} −\dots − q_{r})$ different unordered selections of $n$ objects of $r$ ...
admin
258
views
admin
asked
Apr 30, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
+
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0
votes
0
answers
2018
Kenneth Rosen Edition 7 Exercise 6.1 Question 33 (Page No. 397)
How many strings of eight English letters are there that contain no vowels, if letters can be repeated? that contain no vowels, if letters cannot be repeated? that start with a vowel, if letters can be repeated? that start with a ... be repeated? that start and end with $X$ and contain at least one vowel, if letters can be repeated?
How many strings of eight English letters are therethat contain no vowels, if letters can be repeated?that contain no vowels, if letters cannot be repeated?that start wit...
admin
502
views
admin
asked
Apr 28, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
descriptive
+
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0
votes
0
answers
2019
Kenneth Rosen Edition 7 Exercise 8.1 Question 43 (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 disk is ... then $R(n) = \displaystyle{}\sum_{i = 1}^{k} i2^{i−1} − (t_{k} − n)2^{k−1}.$
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
151
views
admin
asked
May 2, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
descriptive
+
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0
votes
1
answer
2020
Kenneth Rosen Edition 7 Exercise 6.1 Question 36 (Page No. 397)
How many functions are there from the set $\{1, 2,\dots,n\},$ where $n$ is a positive integer, to the set $\{0, 1\}?$
How many functions are there from the set $\{1, 2,\dots,n\},$ where $n$ is a positive integer, to the set $\{0, 1\}?$
admin
349
views
admin
asked
Apr 28, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
descriptive
+
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1
votes
1
answer
2021
JEST 2020
X AND Y is an arbitrary sets, F: $X\rightarrow Y$ show that a and b are equivalent F is one-one For all set Z and function g1: $Z\rightarrow X$ and g2: $Z\rightarrow X$, if $g1 \neq g2$ implies $f \bigcirc g1 \neq f \bigcirc g2$ Where $\bigcirc$ is a fucntion composition.
X AND Y is an arbitrary sets, F: $X\rightarrow Y$ show that a and b are equivalent F is one-oneFor all set Z and function g1: $Z\rightarrow X$ and g2: $Z\rightarrow X$, ...
vivek_mishra
862
views
vivek_mishra
asked
Feb 17, 2020
Set Theory & Algebra
jest
functions
set-theory
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0
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0
answers
2022
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.
admin
164
views
admin
asked
May 1, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
+
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0
votes
0
answers
2023
Kenneth Rosen Edition 7 Exercise 6.5 Question 54 (Page No. 434)
How many ways are there to distribute five indistinguishable objects into three indistinguishable boxes?
How many ways are there to distribute five indistinguishable objects into three indistinguishable boxes?
admin
211
views
admin
asked
May 1, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
combinatory
descriptive
+
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1
votes
2
answers
2024
CMI2016-A-5
A dodecahedron is a regular solid with $12$ faces, each face being a regular pentagon. How many edges are there? And how many vertices? $60$ edges and $20$ vertices $30$ edges and $20$ vertices $60$ edges and $50$ vertices $30$ edges and $50$ vertices
A dodecahedron is a regular solid with $12$ faces, each face being a regular pentagon. How many edges are there? And how many vertices?$60$ edges and $20$ vertices$30$ ed...
go_editor
573
views
go_editor
asked
Dec 30, 2016
Graph Theory
cmi2016
graph-theory
undirected-graph
regular-pentagon
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0
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0
answers
2025
Kenneth Rosen Edition 7 Exercise 6.4 Question 32 (Page No. 422)
Prove the binomial theorem using mathematical induction.
Prove the binomial theorem using mathematical induction.
admin
172
views
admin
asked
Apr 30, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
binomial-theorem
proof
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0
votes
0
answers
2026
Kenneth Rosen Edition 7 Exercise 8.1 Question 37 (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 the historian Flavius Josephus, who was part of a band of $41$ ... $J (100), J (1000),\: \text{and}\: J (10,000)$ from your formula for $J (n).$
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 a...
admin
172
views
admin
asked
May 2, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
descriptive
+
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0
votes
0
answers
2027
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.
admin
194
views
admin
asked
May 1, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
descriptive
+
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0
votes
0
answers
2028
Kenneth Rosen Edition 7 Exercise 6.1 Question 56 (Page No. 398)
The name of a variable in the C programming language is a string that can contain uppercase letters, lowercase letters, digits, or underscores. Further, the first character in the string must be a letter, either uppercase or lowercase ... can be named in C? (Note that the name of a variable may contain fewer than eight characters.)
The name of a variable in the C programming language is a string that can contain uppercase letters, lowercase letters, digits, or underscores. Further, the first charact...
admin
318
views
admin
asked
Apr 28, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
descriptive
+
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0
votes
0
answers
2029
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...
admin
173
views
admin
asked
May 2, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
descriptive
+
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0
votes
0
answers
2030
Kenneth Rosen Edition 7 Exercise 6.1 Question 73 (Page No. 399)
How many diagonals does a convex polygon with $n$ sides have? (Recall that a polygon is convex if every line segment connecting two points in the interior or boundary of the polygon lies entirely within this set and that a diagonal of a polygon is a line segment connecting two vertices that are not adjacent.)
How many diagonals does a convex polygon with $n$ sides have? (Recall that a polygon is convex if every line segment connecting two points in the interior or boundary of ...
admin
220
views
admin
asked
Apr 29, 2020
Combinatory
kenneth-rosen
discrete-mathematics
counting
descriptive
+
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