1.
State precisely the Extended Pigeonhole Principle. Be sure to set up any notation that is required.
Exercises 4.2.3 Exercises
2.
Assume UNCG has \(18502\) students. How many students at UNCG must share a birthday? Explain using the Extended Pigeonhole Principle. (You may assume no one was born on February 29.)
3.
Suppose the final exam is graded on a scale from 0 to 100 points. How many students must be in the class to guarantee that at least two students receive the same score on the final exam?
4.
How many numbers must be selected from the set \(\{1, 3, 5, 7, 9, 11, 13, 15\}\) to guarantee at least one pair of these numbers add up to \(16\text{?}\)
5.
What is the minimum number of people required to be sure that at least four will have birthdays in the same month?
6.
A standard deck of cards consists of \(52\) cards. Each card is one of thirteen ranks (A, 2, 3, \dots, J, Q, K) and one of four suits (\(\clubsuit\text{,}\) \(\spadesuit\text{,}\) \(\heartsuit\text{,}\) \(\diamondsuit\)).
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How many cards must be selected to guarantee that at least three of the same suit are chosen?
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How many cards must be selected to guarantee that at least three of the same rank are selected?
7.
A standard deck of cards consists of \(52\) cards. Each card is one of thirteen ranks (A, 2, 3, \dots, J, Q, K) and one of four suits (\(\clubsuit\text{,}\) \(\spadesuit\text{,}\) \(\heartsuit\text{,}\) \(\diamondsuit\)).
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How many cards must be selected to guarantee that at least two hearts (\(\heartsuit\)) are selected?
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How many cards must be selected to guarantee that at least three spades (\(\spadesuit\)) are selected?
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How many cards must be selected to guarantee that at least two hearts (\(\heartsuit\)) and three spades (\(\spadesuit\)) are selected?
8.
Alice selects clips randomly from a bowl that contains ten large paper clips and thirty small paper clips.
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How many must she select to be sure of having at least three of the same size?
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How many must she select to be sure of having at least five of the same size?
9.
Show that there are at least seventeen people in Greensboro (population \(285000\)) with the same three initials, assuming everyone has a first, middle, and last initial.
10.
There are \(38\) different time periods during which classes can be scheduled. If there are \(650\) different classes, how many different rooms will be needed?
11.
Show that among any group of five integers, there are at least two with the same remainder when divided by \(4\text{.}\)
12.
There are \(50\) baskets of apples. Each basket contains at least one apple and no more than \(24\) apples. Show that there are at least \(3\) baskets containing the same number of apples. If you use the Pigeonhole Principle or its extension, be sure to tell me what are the pigeons and what are the boxes. Hint: The apples are not pigeons. The baskets are not boxes.
