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Recent questions tagged pigeonhole-principle
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31
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?
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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.
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Combinatory
Apr 29, 2020
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admin
279
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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.
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Combinatory
Apr 29, 2020
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admin
439
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kenneth-rosen
discrete-mathematics
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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.
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Combinatory
Apr 29, 2020
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admin
196
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kenneth-rosen
discrete-mathematics
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pigeonhole-principle
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Kenneth Rosen Edition 7 Exercise 6.2 Question 31 (Page No. 406)
Show that there are at least six people in California (population: $37$ million) with the same three initials who were born on the same day of the year (but not necessarily in the same year). Assume that everyone has three initials.
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Combinatory
Apr 29, 2020
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admin
187
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kenneth-rosen
discrete-mathematics
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pigeonhole-principle
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Kenneth Rosen Edition 7 Exercise 6.2 Question 30 (Page No. 406)
Show that if $m$ and $n$ are integers with $m \geq 2 \:\text{and}\: n \geq 2,$ then the Ramsey numbers $R(m, n)\:\text{and}\: R(n, m)$ are equal. $\text{(Recall that Ramsey numbers were discussed after Example}\: 13\: \text{in Section}\: 6.2.)$
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Combinatory
Apr 29, 2020
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admin
228
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kenneth-rosen
discrete-mathematics
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pigeonhole-principle
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Kenneth Rosen Edition 7 Exercise 6.2 Question 29 (Page No. 406)
Show that if $n$ is an integer with $n \geq 2,$ then the Ramsey number $R(2, n)$ equals $n.\text{(Recall that Ramsey numbers were discussed after Example}\: 13\:\text{in Section}\: 6.2.)$
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Combinatory
Apr 29, 2020
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admin
192
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kenneth-rosen
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pigeonhole-principle
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Kenneth Rosen Edition 7 Exercise 6.2 Question 28 (Page No. 406)
Use question $27$ to show that among any group of $20$ people (where any two people are either friends or enemies), there are either four mutual friends or four mutual enemies.
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Combinatory
Apr 29, 2020
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admin
220
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kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 6.2 Question 27 (Page No. 406)
Show that in a group of $10$ people (where any two people are either friends or enemies), there are either three mutual friends or four mutual enemies, and there are either three mutual enemies or four mutual friends.
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Combinatory
Apr 29, 2020
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admin
323
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40
Kenneth Rosen Edition 7 Exercise 6.2 Question 26 (Page No. 406)
Show that in a group of five people (where any two people are either friends or enemies), there are not necessarily three mutual friends or three mutual enemies.
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Combinatory
Apr 29, 2020
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admin
418
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kenneth-rosen
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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.
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Combinatory
Apr 29, 2020
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admin
389
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kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 6.2 Question 24 (Page No. 406)
Suppose that $21$ girls and $21$ boys enter a mathematics competition. Furthermore, suppose that each entrant solves at most six questions, and for every boy-girl pair, there is at least one question that they both solved. Show that there is a question that was solved by at least three girls and at least three boys.
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Combinatory
Apr 29, 2020
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admin
370
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kenneth-rosen
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Kenneth Rosen Edition 7 Exercise 6.2 Question 23 (Page No. 406)
Show that whenever $25$ girls and $25$ boys are seated around a circular table there is always a person both of whose neighbors are boys.
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Combinatory
Apr 29, 2020
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admin
614
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kenneth-rosen
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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.
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Combinatory
Apr 29, 2020
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admin
2.0k
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kenneth-rosen
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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.
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Combinatory
Apr 29, 2020
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admin
1.4k
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kenneth-rosen
discrete-mathematics
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Kenneth Rosen Edition 7 Exercise 6.2 Question 20 (Page No. 406)
Find an increasing subsequence of maximal length and a decreasing subsequence of maximal length in the sequence $22, 5, 7, 2, 23, 10, 15, 21, 3, 17.$
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Combinatory
Apr 29, 2020
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admin
1.8k
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kenneth-rosen
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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
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Combinatory
Apr 29, 2020
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admin
2.5k
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kenneth-rosen
discrete-mathematics
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Kenneth Rosen Edition 7 Exercise 6.2 Question 18 (Page No. 405)
Suppose that there are nine students in a discrete mathematics class at a small college. Show that the class must have at least five male students or at least five female students. Show that the class must have at least three male students or at least seven female students.
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Combinatory
Apr 29, 2020
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admin
4.7k
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kenneth-rosen
discrete-mathematics
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Kenneth Rosen Edition 7 Exercise 6.2 Question 17 (Page No. 405)
A company stores products in a warehouse. Storage bins in this warehouse are specified by their aisle, location in the aisle, and shelf. There are $50$ aisles, $85$ horizontal locations in each aisle, and $5$ shelves ... least number of products the company can have so that at least two products must be stored in the same bin?
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Combinatory
Apr 29, 2020
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admin
3.9k
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kenneth-rosen
discrete-mathematics
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50
Kenneth Rosen Edition 7 Exercise 6.2 Question 16 (Page No. 405)
How many numbers must be selected from the set $\{1, 3, 5, 7, 9, 11, 13, 15\}$ to guarantee that at least one pair of these numbers add up to $16?$
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Combinatory
Apr 29, 2020
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admin
2.2k
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kenneth-rosen
discrete-mathematics
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Kenneth Rosen Edition 7 Exercise 6.2 Question 15 (Page No. 405)
How many numbers must be selected from the set $\{1, 2, 3, 4, 5, 6\}$ to guarantee that at least one pair of these numbers add up to $7?$
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Combinatory
Apr 29, 2020
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admin
722
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kenneth-rosen
discrete-mathematics
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Kenneth Rosen Edition 7 Exercise 6.2 Question 14 (Page No. 405)
Show that if seven integers are selected from the first $10$ positive integers, there must be at least two pairs of these integers with the sum $11.$ Is the conclusion in part $(A)$ true if six integers are selected rather than seven?
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Combinatory
Apr 29, 2020
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admin
2.3k
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kenneth-rosen
discrete-mathematics
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Kenneth Rosen Edition 7 Exercise 6.2 Question 13 (Page No. 405)
Show that if five integers are selected from the first eight positive integers, there must be a pair of these integers with a sum equal to $9.$ Is the conclusion in part $(A)$ true if four integers are selected rather than five?
admin
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Combinatory
Apr 29, 2020
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admin
2.7k
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kenneth-rosen
discrete-mathematics
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Kenneth Rosen Edition 7 Exercise 6.2 Question 12 (Page No. 405)
How many ordered pairs of integers $(a, b)$ are needed to guarantee that there are two ordered pairs $(a_{1}, b_{1})\: \text{and}\: (a_{2}, b_{2})$ such that $a_{1} \mod 5 = a_{2} \mod 5\:\text{and}\: b_{1} \mod 5 = b_{2} \mod 5?$
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Apr 29, 2020
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admin
414
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kenneth-rosen
discrete-mathematics
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pigeonhole-principle
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Kenneth Rosen Edition 7 Exercise 6.2 Question 11 (Page No. 405)
Let $(x_{i}, y_{i}, z_{i}), i = 1, 2, 3, 4, 5, 6, 7, 8, 9,$ be a set of nine distinct points with integer coordinates in $xyz$ space. Show that the midpoint of at least one pair of these points has integer coordinates.
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Combinatory
Apr 29, 2020
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admin
1.2k
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kenneth-rosen
discrete-mathematics
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Kenneth Rosen Edition 7 Exercise 6.2 Question 10 (Page No. 405)
Let $(x_{i}, y_{i}),i = 1, 2, 3, 4, 5,$ be a set of five distinct points with integer coordinates in the $xy$ plane. Show that the midpoint of the line joining at least one pair of these points has integer coordinates.
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Combinatory
Apr 29, 2020
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admin
3.2k
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kenneth-rosen
discrete-mathematics
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Kenneth Rosen Edition 7 Exercise 6.2 Question 9 (Page No. 405)
What is the minimum number of students, each of whom comes from one of the $50$ states, who must be enrolled in a university to guarantee that there are at least $100$ who come from the same state?
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Combinatory
Apr 29, 2020
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admin
6.1k
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kenneth-rosen
discrete-mathematics
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Kenneth Rosen Edition 7 Exercise 6.2 Question 8 (Page No. 405)
Show that if $f$ is a function from $S \:\text{to}\: T ,$ where $S\:\text{and}\: T$ are finite sets with $\mid S\mid > \mid T \mid,$ then there are elements $s_{1}$ and $s_{2}$ in $S$ such that $f(s_{1}) = f (s_{2}),$ or in other words, $f$ is not one-to-one.
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Combinatory
Apr 29, 2020
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admin
705
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kenneth-rosen
discrete-mathematics
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Kenneth Rosen Edition 7 Exercise 6.2 Question 7 (Page No. 405)
Let $n$ be a positive integer. Show that in any set of $n$ consecutive integers there is exactly one divisible by $n.$
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Combinatory
Apr 29, 2020
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admin
2.7k
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kenneth-rosen
discrete-mathematics
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Kenneth Rosen Edition 7 Exercise 6.2 Question 6 (Page No. 405)
Let $d$ be a positive integer. Show that among any group of $d + 1$ (not necessarily consecutive) integers there are two with exactly the same remainder when they are divided by $d.$
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Apr 29, 2020
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admin
1.1k
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