Chapter 6

Find the smallest number of people in a room needed to guarantee that there is a month of the year during which at least five people were born

49

In how many ways can your team win a playoff between two teams if a team wins when they either win two games in a row or win four games?

7

Find the smallest number of people needed in a room to guarantee that at least three were born on the same day of the year (assuming a year has 366 days).

733

Match each description of a set of strings on the left with the number of such strings on the right. 1. Three different uppercase letters followed by two different digits 2. Two different uppercase letters followed by three different digits 3. Four different uppercase letters followed by a digit 4. An uppercase letter followed by four different digits

1. 1,404,000 2. 468,000 3. 3,588,000 4. 131,040

Evaluate each of these r-permutation numbers. 1. P(10, 2) = 2. P(5, 4) = 3. P(8, 1) = 4. P(9, 3) =

1. 90 2. 120 3. 8 4. 504

Put these counting problems in order of their solutions from largest at the top to smallest at the bottom.

1. Number of one-to-one functions from a set with four elements to a set with seven elements 2. Number of different two-letter initials (where the two letters can be the set) 3. Number of bit strings of length ten that start and end with a zero 4. Number of different functions from a set with three elements to a set with six elements

Put the following quantities in order from smallest to largest (with the smallest at the top) where P(n, r) is the number of permutations with r elements from a set with n elements.

1. P(4,3) 2. P(8,2) 3. P(8,3) 4. P(9,3)

Put these counting problems in order of their solutions from largest at the top to smallest at the bottom.Find the number of ways to select a student at a university

1. Peru, Ecuador, Chile, or Argentina if the numbers of students from these countries are 24, 21,18, and 43 respectively. 2. From Mexico, Costa Rica, El Salvador, Nicaragua, or Panama if the numbers of students from these counties are 23, 27, 19, 14, and 22, respectively. 3. From Colorado or California if there are 69 student from Colorado and 35 students from California at the school.

Select all that apply Which of these equals C(11, 4)?

11⋅10⋅9⋅8 / 4! 330 11⋅10⋅9⋅8⋅7⋅6⋅5 / 7! 11! / 4!7!

Find the smallest number of people who live in Canada, a country with 308 federal election districts called ridings, needed to guarantee that there are least five people who live in the same riding.

1233

Find the smallest number of people who live in New Jersey, a state with 21 counties, needed to guarantee that there are least 60 people who live in the same county

1240

Find the smallest number of cars needed so that the license plates of at least four cars begin with the same three characters, each a nonzero digit or uppercase letter.

128626

Find the smallest number of people, each with a first and last name, that we need to select to be sure that at least four have the same two initials.

2029

How many different ways are there to seat five people around a circular table, where two seatings are considered to be the same if each person has the same neighbor to the left and the same neighbor to the right?

24

How many different ways are there to seat five people around a circular table, where two seatings are considered to be the same if each person has the same neighbor to the left and the same neighbor to the right? (Enter your answer as a whole number.)

24

Let C(a, b) denote the number of combinations of b elements from a set with a elements. Which of the following statements are true for all positive integers n and r with 0 ≤ r ≤ n?

C(n + r, n) = C(n + r, r) C(n, n) = 1 C(n, n - r) = C(n, r)

Which of these statements are true of all nonnegative integers n and r with 0 ≤ r ≤ n?

P(n, n) = n! P(n, 1) = n P(n, r) = n! / (n - r)!

Which of these formulae for the cardinality of the union of two sets is correct?

|A ∪ B| = |A| + |B| - |A ∩ B| |A ∪ B| = |A| + |B - A| |A ∪ B| = |A - B| + |B|

Select all that apply For which of these scenarios is it impossible for a function from a set A to a set B to be one-to-one?

|A| = 11, |B| = 10 |A| = 2, |B| = 1

1. How many bit strings of length five either start with two zeros or end with two zeros? 2. How many bit strings of length six either do not start with two zeros or do not end with two zeros? 3. How many bit strings of length seven either start with four ones or end with four ones? 4.How many bit strings of length eight do not start with a one or do not end with a one? 16Blank 4Blank 4 16 , Incorrect Unavailable

1. 14 2. 36 3. 15 4. 192

Match each set of rules for passwords on the left with the number of such passwords on the right. 1. A password consists of five or six characters where each is a digit or a lowercase letter. 2. A password consists of five characters where each is a digit, a lowercase letter, or an uppercase letter or it consists of six characters where each is a digit or a lowercase letter. 3. A password consists of five or six characters where each is a digit or a lowercase letter and there is at least one digit.

1. 2,237,248,512 2. 3,092,915,168 3. 1,916,451,360

1. How many ways are there to select a committee of three people from a department of ten people? 2. How many bit strings of length eight contain exactly four zeros? 3. How many subsets of five elements does a set with seven elements have?

1. 120 2. 70 3. 21

1. Solve these counting problems using the pigeonhole principle.(1) The smallest number of people in a group needed to guarantee that at least two were born in the same month is ______________ 2. .(2) The smallest number of people in a group needed to guarantee that at least two have the same first and last initials is ________________ 3 .(3) The smallest number of people in a group needed to guarantee that at least two have the same first initial and were born on the same day of the week is ____________ 4.(4) The smallest number of people in a group needed to guarantee that at least two were born on the same day of the year, assuming that nobody in the group was born on February 29 in a leap year, is ______________

1. 13 2. 677 3. 183 4. 366

Match the number described on the left with its value on the right. Assume in each case that no race ends in a tie 1. The number of ways to award gold, silver, and bronze medals if there are seven people in a race 2. The number of orders in which six runners can finish a race 3. The number of possibilities for the winner and runner-up in a race with seven runners 4. The number of possibilities for the order of the top four runners in a race with eight runners

1. 210 2. 720 3. 42 4. 1680

Matching Question Match each description of a number on the left with its value on the right. Instructions 1. The number of ways to select four states from the 50 states in the United States 2. The number of ways to select two countries from the 193 member states of the United Nations 3. The number of ways to select six members of the European Union of 28 countries 4. The number of ways to select three of the 101 departments of France

1. 230,300 2. 18,528 3. 376,740 4. 166,650

1.How many different ways are there to identify a room in a large building (a) if each room is identified using an uppercase letter followed by either one nonzero digit or two digits where the first digit is not zero? 2. (b) if each room is identified using a string of one or two digits followed by an uppercase letter? 3. (c) if each room is identified using a string that begins with an uppercase letter, followed by a number between 1 and 50 (inclusive), and some rooms are identified in this same way but end with an extra lowercase letter which is either a, b, or c? 5200Blank 3Blank 3 5200 , Correct Unavailable

1. 2574 2. 2860 3.5200

1. How many elements are in the union of three pairwise disjoint sets if the sets contain 10, 15, and 25 elements? 2. How many ways are there to select a student whose major is in one of the departments of the School of Science if there are seven departments in this school with 31, 88, 19, 11, 41, 22, and 17 students in each? (Assume that no student can have more than one major.) 3. How many ways are there to select a person who lives on a street with five houses if the number of people in these houses are 5, 3, 2, 7, and 6?

1. 50 2. 229 3. 23

Match each counting problem on the left with its answer on the right. 1. Number of bit strings of length nine 2. Number of functions from a set with five elements to a set with four elements 3. Number of one-to-one functions from a set with three elements to a set with eight elements 4. Number of strings of two digits followed by a letter

1. 512 2. 1024 3. 336 4. 2600

1.How many functions are there from a set with four elements to a set with five elements? 2.How many of the functions from a set with four elements to a set with five elements are one-to-one? 3.How many of the functions from a set with four elements to a set with five elements are onto?

1. 5^4 = 625 2. 120 3. 0 or none

1. The number of ways to pick a digit, a lowercase letter, an uppercase letter, or one of eight allowable punctuation marks 2. The number of ways to pick an integer that is between 1 and 50 (inclusive), 100 and 150 (inclusive), or 200 and 250 (inclusive) 3. The number of ways to pick a county in Illinois, Michigan, Minnesota, or Wisconsin if there are 102 counties in Illinois, 83 in Michigan, 87 in Minnesota, and 72 in Wisconsin 4. The number of departments in a university if there are 38 in Literature, Science, and Arts, 13 in Engineering, 17 in Agriculture, and 8 in the Business School

1. 70 2. 150 3. 344 4. 76

1. How many strings can be formed using the letters A, B, C, D, E, F, with each letter used once? 2. How many strings can be formed using the letters A, B, C, D, E, F, with each letter used once, that contain the block CD? 3. How many strings can be formed using the letters A, B, C, D, E, F, with each letter used once, that start with the letter F? 4. How many strings can be formed using the letters A, B, C, D, E, F, with each letter used once, that contain the blocks ABC and EF?

1. 720 2. 120 3. 120 4. 6

Click and drag on elements in order Put the following numbers of combinations in increasing order, with the smallest at the top.

1. C(9,9) 2. C(10,1) 3. C(6,3) 4. C(8,2) 5. C(7,4)

Match each number described on the left with its value on the right, assuming that a drawer contains four white socks and six black socks. 1. The number of socks from the drawer needed to guarantee at least two are white 2. The number of socks from the drawer needed to guarantee at least two are black 3. The number of socks from the drawer needed to guarantee at least two are the same color 4. The number of socks from the drawer needed to guarantee at least one white sock and one black sock

1. Eight 2. Six 3. Three 4. Seven

Match each proof of the identity C(n, r) = C(n, n - r) on the left, where n and r are integers with 0 ≤ r ≤ n, to the type of proof it is on the right. 1. Suppose S is a set with n elements. The function that maps a subset A of S to A⎯⎯⎯is a one-to-one correspondence between subsets of S with r elements and subsets with n - r elements. 2. Suppose S is a set with n elements. The number of subsets of S with relements equals C(n, r). But each subset A of S is also determined by specifying which elements are not in A, and so are in A. Because the complement of a subset of S with relements has n - r elements, there are also C(n, n - r) subsets of S with relements. It follows that C(n, r) = C(n, n - r). 3. C(n, r) = n!r!(n-r)! = n!(n-r)![n-(n-r)]! = C(n, n - r)

1. Proof of Bijectivity 2. Double Counting Proof 3. Algebraic Proof

Ordering Question Click and drag on elements in order Put these counting problems in increasing order of their solutions, with the smallest at the top.

1. The number of different committees of 4 people that can be formed form a department with 12 members 2. The number of different committees of 3 people that can be formed from a department with 16 menbers 3. The number of different committees of 5 people that can be formed by selecting 3 people from a department with 11 members and 2 from one with 12 4. The number of different committees of 6 people that can be formed by selecting 2 people from a department with 13 members and 4 from one with 10

Which of these types of proofs can be used to prove a combinatorial identity?

A double counting argument that shows the expressions on the left side and on the right side count the same objects, but in different ways. A proof showing that there is a bijection between objects counted by the left side and objects counted by the right side. An algebraic proof that uses the rules of algebra to show that the expressions on the two sides of the identity are equal.

Which of these numbers of different license plates can be determined using only the product rule (without the use of the sum rule or some other rule)? (Select all that apply.)

The number of license plates consisting of a sequence of three uppercase letters followed by three digits The number of license plates consisting of a sequence of six uppercase letters