Question 1

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

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Question 2

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 apart.

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Question 3

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 4

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

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Question 5

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

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Question 6

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

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Question 7

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 8

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

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Question 9

The ﬁgure at the right shows a 4-block by 5-block grid of streets.
to point B, where at each stage you can only go right or up.
Find the number of ways in which you can go from point A  (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 10

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 11

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 12

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 13

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain neither 5 nor 6.

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Question 14

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 15

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 16

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 17

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."

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Question 18

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

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Question 19

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 20

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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Quiz 6: Counting

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

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 apart.

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

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

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.

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

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

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} that contain exactly three elements, one of which is 3.

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain neither 5 nor 6.

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

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.

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.

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."

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

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