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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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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?
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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 $0s\: \text{and}\: 1s?$
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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 $1s?$
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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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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?
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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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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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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)$$C(12, 6)$
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Kenneth Rosen Edition 7 Exercise 6.3 Question 5 (Page No. 413)
Find the value of each of these quantities. $P (6, 3)$ $P (6, 5)$ $P (8, 1)$ $P (8, 5)$ $P (8, 8)$ $P (10, 9)$
Find the value of each of these quantities.$P (6, 3)$$P (6, 5)$$P (8, 1)$$P (8, 5)$$P (8, 8)$$P (10, 9)$
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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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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 $a?$
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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 $\{a, b, c, d, e, f, g\}?$
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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 $\{a, b, c\}.$
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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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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.
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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...
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Kenneth Rosen Edition 7 Exercise 6.2 Question 22 (Page No. 406)
Show that if there are $101$ people of different heights standing in a line, it is possible to find $11$ people in the order they are standing in the line with heights that are either increasing or decreasing.
Show that if there are $101$ people of different heights standing in a line, it is possible to find $11$ people in the order they are standing in the line with heights th...
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Kenneth Rosen Edition 7 Exercise 6.2 Question 21 (Page No. 406)
Construct a sequence of $16$ positive integers that has no increasing or decreasing subsequence of five terms.
Construct a sequence of $16$ positive integers that has no increasing or decreasing subsequence of five terms.
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Kenneth Rosen Edition 7 Exercise 6.2 Question 19 (Page No. 405 - 406)
Suppose that every student in a discrete mathematics class of $25$ students is a freshman, a sophomore, or a junior. Show that there are at least nine freshmen, at least nine sophomores, or at least nine juniors in the ... there are either at least three freshmen, at least $19$ sophomores, or at least five juniors in the class
Suppose that every student in a discrete mathematics class of $25$ students is a freshman, a sophomore, or a junior.Show that there are at least nine freshmen, at least n...
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