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Kenneth Rosen Edition 7 Exercise 6.3 Question 14 (Page No. 413)
In how many ways can a set of two positive integers less than $100$ be chosen?
In how many ways can a set of two positive integers less than $100$ be chosen?
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Apr 29, 2020
Combinatory
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452
Kenneth Rosen Edition 7 Exercise 6.3 Question 13 (Page No. 413)
A group contains $n$ men and $n$ women. How many ways are there to arrange these people in a row if the men and women alternate?
A group contains $n$ men and $n$ women. How many ways are there to arrange these people in a row if the men and women alternate?
admin
510
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admin
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Apr 29, 2020
Combinatory
kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 6.3 Question 12 (Page No. 413)
How many bit strings of length $12$ contain exactly three $1s?$ at most three $1s?$ at least three $1s?$ an equal number of $0s\: \text{and}\: 1s?$
How many bit strings of length $12$ containexactly three $1s?$ at most three $1s?$ at least three $1s?$ an equal number of $0s\: \text{and}\: 1s?$
admin
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Apr 29, 2020
Combinatory
kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 6.3 Question 11 (Page No. 413)
How many bit strings of length $10$ contain exactly four $1s?$ at most four $1s?$ at least four $1s?$ an equal number of $0s$ and $1s?$
How many bit strings of length $10$ containexactly four $1s?$at most four $1s?$at least four $1s?$an equal number of $0s$ and $1s?$
admin
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Apr 29, 2020
Combinatory
kenneth-rosen
discrete-mathematics
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Kenneth Rosen Edition 7 Exercise 6.3 Question 10 (Page No. 413)
There are six different candidates for governor of a state. In how many different orders can the names of the candidates be printed on a ballot?
There are six different candidates for governor of a state. In how many different orders can the names of the candidates be printed on a ballot?
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Apr 29, 2020
Combinatory
kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 6.3 Question 9 (Page No. 413)
How many possibilities are there for the win, place, and show (first, second, and third) positions in a horse race with $12$ horses if all orders of finish are possible?
How many possibilities are there for the win, place, and show (first, second, and third) positions in a horse race with $12$ horses if all orders of finish are possible?
admin
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Apr 29, 2020
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457
Kenneth Rosen Edition 7 Exercise 6.3 Question 8 (Page No. 413)
In how many different orders can five runners finish a race if no ties are allowed?
In how many different orders can five runners finish a race if no ties are allowed?
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350
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Apr 29, 2020
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Kenneth Rosen Edition 7 Exercise 6.3 Question 7 (Page No. 413)
Find the number of $5$-permutations of a set with nine elements.
Find the number of $5$-permutations of a set with nine elements.
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Apr 29, 2020
Combinatory
kenneth-rosen
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459
Kenneth Rosen Edition 7 Exercise 6.3 Question 6 (Page No. 413)
Find the value of each of these quantities. $C(5, 1)$ $C(5, 3)$ $C(8, 4)$ $C(8, 8)$ $C(8, 0)$ $C(12, 6)$
Find the value of each of these quantities.$C(5, 1)$$C(5, 3)$$C(8, 4)$$C(8, 8)$$C(8, 0)$$C(12, 6)$
admin
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Apr 29, 2020
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kenneth-rosen
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460
Kenneth Rosen Edition 7 Exercise 6.3 Question 5 (Page No. 413)
Find the value of each of these quantities. $P (6, 3)$ $P (6, 5)$ $P (8, 1)$ $P (8, 5)$ $P (8, 8)$ $P (10, 9)$
Find the value of each of these quantities.$P (6, 3)$$P (6, 5)$$P (8, 1)$$P (8, 5)$$P (8, 8)$$P (10, 9)$
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Apr 29, 2020
Combinatory
kenneth-rosen
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461
Kenneth Rosen Edition 7 Exercise 6.3 Question 4 (Page No. 413)
Let $S = \{1, 2, 3, 4, 5\}.$ List all the $3$-permutations of $S$. List all the $3$-combinations of $S.$
Let $S = \{1, 2, 3, 4, 5\}.$List all the $3$-permutations of $S$.List all the $3$-combinations of $S.$
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Apr 29, 2020
Combinatory
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Kenneth Rosen Edition 7 Exercise 6.3 Question 3 (Page No. 413)
How many permutations of $\{a, b, c, d, e, f, g\}$ end with $a?$
How many permutations of $\{a, b, c, d, e, f, g\}$ end with $a?$
admin
279
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Apr 29, 2020
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kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 6.3 Question 2 (Page No. 413)
How many different permutations are there of the set $\{a, b, c, d, e, f, g\}?$
How many different permutations are there of the set $\{a, b, c, d, e, f, g\}?$
admin
258
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Apr 29, 2020
Combinatory
kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 6.3 Question 1 (Page No. 413)
List all the permutations of $\{a, b, c\}.$
List all the permutations of $\{a, b, c\}.$
admin
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Apr 29, 2020
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kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 6.2 Question 47 (Page No. 407)
An alternative proof of Theorem $3$ ... there is no increasing subsequence of length $n + 1,$ then there must be a decreasing subsequence of this length.
An alternative proof of Theorem $3$ based on the generalized pigeonhole principle is outlined in this exercise. The notation used is the same as that used in the proof in...
admin
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Apr 29, 2020
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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
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Apr 29, 2020
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Kenneth Rosen Edition 7 Exercise 6.2 Question 45 (Page No. 407)
Let $x$ be an irrational number. Show that for some positive integer $j$ not exceeding the positive integer $n,$ the absolute value of the difference between $j x$ and the nearest integer to $j x$ is less than $1/n.$
Let $x$ be an irrational number. Show that for some positive integer $j$ not exceeding the positive integer $n,$ the absolute value of the difference between $j x$ and th...
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430
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Apr 29, 2020
Combinatory
kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 6.2 Question 44 (Page No. 406)
There are $51$ houses on a street. Each house has an address between $1000\: \text{and}\: 1099,$ inclusive. Show that at least two houses have addresses that are consecutive integers.
There are $51$ houses on a street. Each house has an address between $1000\: \text{and}\: 1099,$ inclusive. Show that at least two houses have addresses that are consecut...
admin
450
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Apr 29, 2020
Combinatory
kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 6.2 Question 43 (Page No. 406)
Show that if $f$ is a function from $S\: \text{to}\: T,$ where $S\: \text{and}\: T$ are nonempty finite sets and $m = \left \lceil \mid S \mid / \mid T \mid \right \rceil ,$ then there are at least $m$ elements of $S$ mapped to the same value ... $f (s_{1}) = f (s_{2}) =\dots = f (s_{m}).$
Show that if $f$ is a function from $S\: \text{to}\: T,$ where $S\: \text{and}\: T$ are nonempty finite sets and $m = \left \lceil \mid S \mid / \mid T \mid \right \r...
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Apr 29, 2020
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470
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
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Apr 29, 2020
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Kenneth Rosen Edition 7 Exercise 6.2 Question 41 (Page No. 406)
An arm wrestler is the champion for a period of $75$ hours. (Here, by an hour, we mean a period starting from an exact hour, such as $1\: \text{p.m.,}$ until the next hour.) The arm wrestler had at ... than $125$ total matches. Show that there is a period of consecutive hours during which the arm wrestler had exactly $24$ matches.
An arm wrestler is the champion for a period of $75$ hours. (Here, by an hour, we mean a period starting from an exact hour, such as $1\: \text{p.m.,}$ until the next hou...
admin
275
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Apr 29, 2020
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kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 6.2 Question 40 (Page No. 406)
Prove that at a party where there are at least two people, there are two people who know the same number of other people there.
Prove that at a party where there are at least two people, there are two people who know the same number of other people there.
admin
481
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Apr 29, 2020
Combinatory
kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 6.2 Question 39 (Page No. 406)
Find the least number of cables required to connect $100$ computers to $20$ printers to guarantee that $2$ every subset of $20 $computers can directly access $20$ different printers. (Here, the assumptions about cables and computers are the same as in Example $9.$) Justify your answer.
Find the least number of cables required to connect $100$ computers to $20$ printers to guarantee that $2$ every subset of $20 $computers can directly access $20$ differe...
admin
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Apr 29, 2020
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kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 6.2 Question 38 (Page No. 406)
Find the least number of cables required to connect eight computers to four printers to guarantee that for every choice of four of the eight computers, these four computers can directly access four different printers. Justify your answer.
Find the least number of cables required to connect eight computers to four printers to guarantee that for every choice of four of the eight computers, these four compute...
admin
477
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admin
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Apr 29, 2020
Combinatory
kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 6.2 Question 37 (Page No. 406)
A computer network consists of six computers. Each computer is directly connected to zero or more of the other computers. Show that there are at least two computers in the network that are directly connected to the same number of ... is impossible to have a computer linked to none of the others and a computer linked to all the others.]
A computer network consists of six computers. Each computer is directly connected to zero or more of the other computers. Show that there are at least two computers in th...
admin
326
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admin
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Apr 29, 2020
Combinatory
kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 6.2 Question 36 (Page No. 406)
A computer network consists of six computers. Each computer is directly connected to at least one of the other computers. Show that there are at least two computers in the network that are directly connected to the same number of other computers.
A computer network consists of six computers. Each computer is directly connected to at least one of the other computers. Show that there are at least two computers in th...
admin
233
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Apr 29, 2020
Combinatory
kenneth-rosen
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477
Kenneth Rosen Edition 7 Exercise 6.2 Question 35 (Page No. 406)
There are $38$ different time periods during which classes at a university can be scheduled. If there are $677$ different classes, how many different rooms will be needed?
There are $38$ different time periods during which classes at a university can be scheduled. If there are $677$ different classes, how many different rooms will be needed...
admin
495
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admin
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Apr 29, 2020
Combinatory
kenneth-rosen
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478
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
289
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Apr 29, 2020
Combinatory
kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 6.2 Question 33 (Page No. 406)
In the $17^{\text{th}} $ century, there were more than $800,000$ inhabitants of Paris. At the time, it was believed that no one had more than $200,000$ hairs on their head. Assuming these numbers are correct and that ... to show that there had to be at least five Parisians at that time with the same number of hairs on their heads.
In the $17^{\text{th}} $ century, there were more than $800,000$ inhabitants of Paris. At the time, it was believed that no one had more than $200,000$ hairs on their hea...
admin
456
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admin
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Apr 29, 2020
Combinatory
kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 6.2 Question 32 (Page No. 406)
Show that if there are $100,000,000$ wage earners in the United States who earn less than $1,000,000$ dollars (but at least a penny), then there are two who earned exactly the same amount of money, to the penny, last year.
Show that if there are $100,000,000$ wage earners in the United States who earn less than $1,000,000$ dollars (but at least a penny), then there are two who earned exactl...
admin
200
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Apr 29, 2020
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