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3242
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 $16?$
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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3243
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 $7?$
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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3249
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?
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 $100$ wh...
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3251
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 $n.$
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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3252
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 $d.$
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 div...
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3253
Kenneth Rosen Edition 7 Exercise 6.2 Question 5 (Page No. 405)
Show that among any group of five (not necessarily consecutive) integers, there are two with the same remainder when divided by $4.$
Show that among any group of five (not necessarily consecutive) integers, there are two with the same remainder when divided by $4.$
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3256
Kenneth Rosen Edition 7 Exercise 6.2 Question 2 (Page No. 405)
Show that if there are $30$ students in a class, then at least two have last names that begin with the same letter.
Show that if there are $30$ students in a class, then at least two have last names that begin with the same letter.
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3257
Kenneth Rosen Edition 7 Exercise 6.2 Question 1 (Page No. 405)
Show that in any set of six classes, each meeting regularly once a week on a particular day of the week, there must be two that meet on the same day, assuming that no classes are held on weekends.
Show that in any set of six classes, each meeting regularly once a week on a particular day of the week, there must be two that meet on the same day, assuming that no cla...
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3258
Kenneth Rosen Edition 7 Exercise 6.1 Question 65 (Page No. 398)
How many ways are there to arrange the letters $a, b, c,\:\text{and}\: d$ such that $a$ is not followed immediately by $b?$
How many ways are there to arrange the letters $a, b, c,\:\text{and}\: d$ such that $a$ is not followed immediately by $b?$
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3259
Kenneth Rosen Edition 7 Exercise 6.1 Question 64 (Page No. 398)
Use a tree diagram to find the number of bit strings of length four with no three consecutive $0s.$
Use a tree diagram to find the number of bit strings of length four with no three consecutive $0s.$
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3260
Kenneth Rosen Edition 7 Exercise 6.1 Question 63 (Page No. 398)
Use the principle of inclusion–exclusion to find the number of positive integers less than $1,000,000$ that are not divisible by either $4\:\text{or}\: \text{by}\: 6.$
Use the principle of inclusion–exclusion to find the number of positive integers less than $1,000,000$ that are not divisible by either $4\:\text{or}\: \text{by}\: 6.$
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