Discrete Math Test 2

2⁸

How many bit strings of length 10 both begin and end with 1?

a. C(13,6) = 1716 b. C(19,12) = 50,388 c. C(31,24) = 2,629,575 d. C(11,4) = 330

A bagel shop has onion, poppy, egg, salty, pumpernickel, sesame, raisin, and plain bagels. How many ways are there to choose a. 6 bagels? b. a dozen bagels? c. two dozen bagels? d. a dozen bagels with at least one of each kind?

C(3002,3000) = 4,504,501

A book publisher has 3000 copies of a discrete mathematics book. How many ways are there to store these books in their three warehouses if the copies of the book are indistinguishable?

a. C(25,4) = 12,650 b. 25×24×23×22 = 303,600

A club has 25 members. a. How many ways are there to choose four members of the club to serve on an executive committee? b. How many ways are there to choose a president, vice president, secretary, and treasurer of the club, where no person can hold more than one office?

a. 2¹⁰ = 1024 b. C(10,3) = 120 c. 2¹⁰ - [C(10,2)+C(10,1)+C(10,0)] = 968 d. C(10,5) = 252

A coin is flipped 10 times where each flip comes up either heads or tails. How many possible outcomes a. are there in total? b. contain exactly 3 heads? c. contain at least 3 heads? d. contain the same number of heads and tails?

3⁵⁰

A committee is formed consisting of one representative from each of the 50 states in the United States, where the representative from a state is either the governor or one of the two senators from that state. How many ways are there to form this committee?

2×n!×n! = 2(n!)²

A group contains n men and n women. How many ways are there to arrange these people in a row if the men and women alternate?

a. 4¹⁰ b. 5¹⁰

A multiple-choice test contains 10 questions. There are four possible answers for each question. a. In how many ways can a student answer the questions on the test if the student answers every question? b. In how many ways can a student answer the questions on the test if the student can leave answers blank?

C(4,0)x⁴ + C(4,1)x³y + C(4,2)x²y² + C(4,3)xy³ + C(4,4)y⁴ = x⁴+4x³y+6x²y²+4xy³+y⁴

Find the expansion of (x+y)⁴

P(9,5) = 15,120

Find the number of 5-permutations of a set with nine elements

a. 5!/(5-1)! = 5 b. 6! = 720 c. 8!÷(8-1)! = 8 d. 8!÷(8-5)! = 6720 e. 8!÷(8-8)! = 40,320 f. 10!÷(10-9)! = 3,628,800

Find the value of each of these quantities a. P(5,1) b. P(6, 5) c. P(8, 1) d. P(8, 5) e. P(8, 8) f. P(10, 9)

a. C(10,4) = 210 b. C(10,4)+C(10,3)+C(10,2)+C(10,1)+C(10,0) = 386 c. C(10,4)+C(10,5)+C(10,6)+C(10,7)+C(10,8)+ C(10,9)+C(10,10) = 848 d. C(10,5) = 252

How many bit strings of length 10 contain a. exactly four 1s? b. at most four 1s? c. at least four 1s? d. an equal number of 0s and 1s?

C(10,3)+C(10,4)+C(10,5)+C(10,6)+c(10,7) = 912

How many bit strings of length 10 contain at least 3 1s and at least 3 0s?

(2⁸+2⁷)-2⁵ = 352

How many bit strings of length 10 either begin with three 0s or end with two 0s?

n

How many bit strings with length not exceeding n, where n is a positive integer, consist entirely of 1s, not counting the empty string?

11!÷[5!2!2!] = 83,160

How many different strings can be made from the letters in ABRACADABRA, using all the letters?

26² = 676

How many different three-letter initials are there that begin with A?

26³

How many different three-letter initials can people have?

[26²×10²]+[26²×10³]+[26³×10²]+[26³×10³] = 20,077,200

How many license plates can be made using either two or three uppercase English letters followed by either two or three digits?

[26²×10⁴]+[10²×26⁴] = 52,457,600

How many license plates can be made using either two uppercase English letters followed by four digits or two digits followed by four uppercase English letters?

26×25×24×10×9×8 = 11,232,000

How many license plates consisting of three letters followed by three digits contain no letter or digit twice?

a. 7! = 5040 b. 6! = 720 c. 5! = 120 d. 5! = 120 e. 4! = 24 (CAB and BED both contain B therefore CABED must be a substring) f. 0 (impossible since BCA and ABF both contain B)

How many permutations of the letters ABCDEFGH contain a. the string ED? b. the string CDE? c. the strings BA and FGH? d. the strings AB, DE, GH? e. the strings CAB and BED? f. the strings BCA and ABF?

6! = 720

How many permutations of {abcdefg} end with a?

12×11×10 = 1320

How many possibilities are there for the win, place, and show (first, second, and third) positions in a horse race with 12 horses if all orders of finish are possible?

a. C(24,20) = 10,626 b. C(15,11) = 1365 c. C(25,21)-C(24,21)-C(14,10) = 9625

How many solutions are there to the equation x₁+x₂+x₃+x₄+x₅ = 21 where x is a nonnegative integer such that a. x₁ ≥ 1? b. all x ≥ 2 c. 0 ≤ x₁ ≤ 10

26+26²+26³+26⁴ = 475,254

How many strings are there of lowercase letters of length four or less, not counting the empty string?

26⁶ = 308,915,776

How many strings of 6 letters are there?

a. 5×21⁵×C(6,1) = 122,523, 030 b. (5²×21⁴)×C(6,2) = 72,930,375 c. 26⁶-21⁶ = 223,149,665 d. 26⁶-21⁶-122,523,030 = 100,626,625

How many strings of 6 lowercase letters of the English alphabet contain a. exactly one vowel? b. exactly two vowels? c. at least one vowel? d. at least two vowels?

a. 21⁸ = 37,822,859,361 b. 21×20×...×14 = 8,204,716,800 c. 5×26⁷ = 40,159,050,880 d. 5×25×24×...×19 = 12,113,640,000 e. 26⁸-21⁸ = 171,004,205,215 f. 8×(5×21⁷) = 72,043,541,640 g. 26⁷-21⁷ = 6,230,721,635 h. 26⁶-21⁶ = 223,149,655

How many strings of eight English letters are there a. that contain no vowels, if letters can be repeated? b. that contain no vowels, if letters cannot be repeated? c. that start with a vowel, if letters can be repeated? d. that start with a vowel, if letters cannot be repeated? e. that contain at least one vowel, if letters can be repeated? f. that contain exactly one vowel, if letters can be repeated? g. that start with X and contain at least one vowel, if letters can be repeated? h. that start and end with X and contain at least one vowel, if letters can be repeated?

a. 10³-10 = 990 b. 5×10² = 500 c. 10+10+10-3 = 27

How many strings of three decimal digits a. do not contain the same digit three times? b. begin with an odd digit? c. have exactly 2 digits that are 4?

2¹⁰⁰- C(100,0) -C(100,1)-C(100,2) = 2¹⁰⁰-5051

How many subsets with more than two elements does a set with 100 elements have?

101

How many terms are there in the expansion of (x+y)¹⁰⁰?

8!×C(9,5)×5! = 609,638,400

How many ways are there for 8 men and 5 women to stand in a line so that no two women stand next to each other?

5³ = 125

How many ways are there to assign three jobs to five employees if each employee can be given more than one job?

9 Simply count: 0 pennies and 8 nickels 1 penny and 7 nickels 2 pennies and 6 nickels 3 pennies and 5 nickels 4 pennies and 4 nickels 5 pennies and 3 nickels 6 pennies and 2 nickels 7 pennies and 1 nickel 8 pennies and 0 nickels

How many ways are there to choose eight coins from a piggy bank containing 100 identical pennies and 80 identical nickels?

C(7,3) = 35 (3 stars, 4 bars)

How many ways are there to select three unordered elements from a set with five elements when repetition is allowed?

3⁵ = 243

In how many different ways can five elements be selected in order from a set with three elements when repetition is allowed?

a. 2×5! = 240 b. 6!-(2×5!) = 480 c. 6!÷2 = 360

In how many ways can a photographer at a wedding arrange six people in a row, including the bride and groom, if a. the bride must be next to the groom? b. the bride is not next to the groom? c. the bride is positioned somewhere to the left of the groom?

C(26,5) = 65,780

In how many ways can a set of five letters be selected from the English alphabet?

{{abc}, {acb}, {bac}, {bca}, {cab}, {cba}}

List all of the permutations of {abc}

a. 100×99×98×97 = 94,109,400 b. 99×98×97 = 941,094 c. 4×(99×98×97) = 3,764,376 d. 99×98×97×96 = 90,345,024 e. 98×97×P(4,2) = 114,072 f. 97×P(4,3) = 2328 g. 4! = 24 h. 96×95×94×93 = 79,727,040 i. 4×99×98×97 = 3,764,376 j. 96×95×P(4,2) = 109,440

One hundred tickets, numbered 1,2,3,...,100, are sold to 100 different people for a drawing. Four different prizes are awarded, including a grand prize (a trip to Tahiti). How many ways are there to award the prizes if a. there are no restrictions? b. the person holding ticket 47 wins the grand prize? c. the person holding ticket 47 wins one of the prizes? d. the person holding ticket 47 does not win a prize? e. the people holding tickets 19 and 47 both win prizes? f. the people holding tickets 19, 47, and 73 all win prizes? g. the people holding tickets 19, 47, 73, and 97 all win prizes? h. none of the people holding tickets 19, 47, 73, and 97 win prizes? i. the grand prize winner is a person holding ticket 19, 47, 73, or 97? j. the people holding tickets 19 and 47 win prizes, but the people holding tickets 73 and 97 do not win prizes?

7×6 = 42

Six different flights fly from New York to Denver and seven fly from Denver to San Francisco. How many different pairs of airlines can you choose on which to book a trip from New York to San Francisco via Denver, when you pick an airline for the flight to Denver and an airline for the continuation flight to San Francisco?

C(15,3)×C(10,3) = 54,600

Suppose that a department contains 10 men and 15 women. How many ways are there to form a committee with six members if it must have the same number of men and women?

a. 325×18 = 5850 b. 325+18 = 343

There are 18 mathematics majors and 325 computer science majors at a college. a. In how many ways can two representatives be picked so that one is a mathematics major and the other is a computer science major? b. In how many ways can one representative be picked who is either a mathematics major or a computer science major?

-2¹⁰×C(19,10)

What is the coefficient of x⁹ in (2-x)¹⁹