Question 1

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly ﬁve elements, the sum of which is even.

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Question 2

Suppose a restaurant serves a "special dinner" consisting of soup, salad, entree, dessert, and beverage. The restaurant has ﬁve kinds of soup, three kinds of salad, ten entrees, ﬁve desserts, and four beverages. How many diﬀerent special dinners are possible? (Two special dinners are diﬀerent if they diﬀer in at least one selection.)

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Question 3

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly three elements.

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Question 4

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly four elements, the sum of which is even.

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Question 5

Show that if five points are picked on or in the interior of a square of side length 2 , then there are at least two of these points no farther than $\sqrt { 2 }$ apart.

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Question 6

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly four elements, the sum of which is odd.

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Question 7

A factory makes automobile parts. Each part has a code consisting of a digit, a letter, and a digit, with the digits distinct, such as 5C7, 1O6, or 3Z0. Last week the factory made 5,000 parts. Find the minimum number of parts that must have the same serial number.

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Question 8

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain both 5 and 6.

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Question 9

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly three elements, one of which is 3.

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Question 10

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly three elements, all of them even.

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Question 11

A game consisting of ﬂipping a coin ends when the player gets two heads in a row, two tails in a row, or ﬂips the coin four times. (a) Draw a tree diagram to show the ways in which the game can end. (b) In how many ways can the game end?

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Question 12

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly ﬁve elements, all of them even.

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Question 13

Find the number of subsets of S = {1, 2, 3, . . . , 10} with exactly ﬁve elements, two of which are 3 and 4.

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Question 14

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain no odd numbers.

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Question 15

A professor teaching a Discrete Math course gives a multiple choice quiz that has ten questions, each with four possible responses: a, b, c, d. What is the minimum number of students that must be in the professor's class in order to guarantee that at least three answer sheets must be identical? (Assume that no answers are left blank.)

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Question 16

The figure at the right shows a 4-block by 5-block grid of streets. Find the number of ways in which you can go from point  A  to point  B , where at each stage you can only go right or up. (You are not allowed to go left or down.) For example, one allowable route from  A  to  B  is: Right, Right, Up, Right, Up, Up, Right, Right, Up.

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Question 17

Find the number of subsets of S = {1, 2, 3, . . . , 10} with exactly ﬁve elements, including 3 or 4 but not both.

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Question 18

A factory makes automobile parts. Each part has a code consisting of a letter and three digits, such as C117, O076, or Z920. Last week the factory made 60,000 parts. Prove that there are at least three parts that have the same serial number.

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Question 19

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly ﬁve elements, but neither 3 nor 4.

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Question 20

Here is an incorrect solution to a problem. Find the error, explain why it is not correct, and give the correct answer. "Problem: Find the number of ways to get two pairs of two different ranks (such as 2 jacks and 2 fives) in a 4-card hand from an ordinary deck of 52 cards."
"Solution: There are 13 ways to get a rank (such as "kings") for the first pair and ( $\left. \begin{array} { l } 4 \ 2 \end{array} \right)$ ways to get a pair of that rank. Similarly, there are 12 ways to get a rank (such as "sevens") for the second pair and $\left( \begin{array} { l } 4 \ 2 \end{array} \right)$ ways to get a pair of that rank. Therefore there are $13 \cdot \left( \begin{array} { l } 4 \ 2 \end{array} \right) \cdot 12 \cdot \left( \begin{array} { l } 4 \ 2 \end{array} \right)$ ways to get 2 pairs."

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Quiz 6: A: Counting

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly ﬁve elements, the sum of which is even.

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly three elements.

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly four elements, the sum of which is even.

Show that if five points are picked on or in the interior of a square of side length 2 , then there are at least two of these points no farther than 2\sqrt { 2 }2​ apart.

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly four elements, the sum of which is odd.

A factory makes automobile parts. Each part has a code consisting of a digit, a letter, and a digit, with the digits distinct, such as 5C7, 1O6, or 3Z0. Last week the factory made 5,000 parts. Find the minimum number of parts that must have the same serial number.

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain both 5 and 6.

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly three elements, one of which is 3.

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly three elements, all of them even.

A game consisting of ﬂipping a coin ends when the player gets two heads in a row, two tails in a row, or ﬂips the coin four times. (a) Draw a tree diagram to show the ways in which the game can end. (b) In how many ways can the game end?

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly ﬁve elements, all of them even.

Find the number of subsets of S = {1, 2, 3, . . . , 10} with exactly ﬁve elements, two of which are 3 and 4.

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain no odd numbers.

Find the number of subsets of S = {1, 2, 3, . . . , 10} with exactly ﬁve elements, including 3 or 4 but not both.

A factory makes automobile parts. Each part has a code consisting of a letter and three digits, such as C117, O076, or Z920. Last week the factory made 60,000 parts. Prove that there are at least three parts that have the same serial number.

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly ﬁve elements, but neither 3 nor 4.

Show that if five points are picked on or in the interior of a square of side length 2 , then there are at least two of these points no farther than $\sqrt { 2 }$ apart.