CSC 226 Test 1

Suppose that a department contains 10 men and 15 women. How many ways are there to form a committee with 6 members if it must have more women than men?

96460

a ) A gallery has 5 painting by artist A, 6 paintings by artist B, and 3 paintings by artist C. In how many ways can they be placed across a wall? b) How many ways if paintings by the same artist must be together?

A) 14! B) 3! * 5! * 6! * 3!

Show that there are at least four people in California (population: 25 million ) with the same three initials who were born on the same day of the year (but no necessarily in the same year). Note: For this problem assume 366 days in a year A) Number of pigeons = B) Number of Holes = C) Equation? D) # of different initial & day of the year combinations possible

A) 25000000 B) 6432816 C) (25000000/6432816) = 4 D) 6432816

Show that in any set of six classes there must be two that meet on the same day, assuming that no classes are held on weekends. A) Number of pigeons = B) Number of Holes = C) Equation?

A) 6 B) 5 C) 6/5 = 2

A) Given a soccer team of 23 players, how many di↵erent possible starting 11 combinations are there? B) What if we know the 23 consists of 3 goalkeeper, 7 defenders, 6 midfielders, and 7 forwards and the starting 11 will have 1 goalkeeper, 4 defenders, 3 midfielders, and 3 forwards?

A) C (23,11) B) C(3,1) * C(7,4) * C(6,3) * C(7,3)

A) There are 11 programs queued to run on a quad-core processor (processor with four cores). If each program can only be assigned to at most one core, and each core will have one program running on it, how many different combinations of programs can be run at the same time? B) If any of the programs can be run on multiple cores at the same time, how many different assignments are there now?

A) C(11,4) B) 1001

How many binary numbers are there with a length between three and five (inclusively)?

2^3 + 2^4 + 2^5

If we are ordering ice cream sundaes, we have three ice cream flavor options and four topping options. Each sundae is one ice cream flavor and 0-4 toppings. How many ways to choose toppings after flavor is chosen?

2^4

A system encodes character using 8 bits, but the first 4 bits are reserved for future use (currently always 0). How many di↵erent strings can be represented with 32 bits?

2^4 * 2^4 * 2^4 * 2^4 = 2^16

How many 8-bit strings are there that do not begin with 000?

2^8 - 2^5

How many strings are there over the set a, b, c that have length 10 in which no two consecutive characters are the same? For example, the string "abcbcbabcb" would count and the strings "abbbcbabcb" and "aacbcbabcb" would not count.

3 * 2^9

How many binary strings of length 5 are there?

32

How many passwords with a length between six and eight (inclusively) can be created with numbers and uppercase letters?

36^6 + 36^7 + 36^8

How many passwords of length 6 are there that contain at least one letter and at least one number (0-9)? You may assume that passwords can only contain uppercase letters and numbers.

36^6 - 26^6 - 10^6

How many permutations of 3 elements from {a, b, c, d, e}?

5 * 4 * 3

In how many different orders can five runners finish a race?

5*4*3*2*1

How many license plates can be made using either 3 letters followed by 3 digits or 4 letters followed by 2 digits? Numerical answer =

63273600

Assuming we have three WolfLine Buses A, B and C, and buses A and B each have a maximum capacity of 20 students. What is the minimum number of students required to ensure that one of these buses has at least 26 students in it?

66

Assuming we have three WolfLine Busses A, B and C. What is the minimum number of students required to ensure that at least one of these buses has at least 26 students in it?

76

How many 8-bit strings are there that start with 1 or end in 00?

(2^7 + 2^6) - 2^5

Tiffany lives alone. (use d)

-Ed L(Tiffany,d)

How many distinct strings can be made by rearranging COMPETITIVE letters, such that 'P' comes immediately before 'O'?

10!/[2!2!2!]

How many distinct strings can be made by rearranging COMPETITIVE letters, such that 'P' comes before 'O'?

11!/[2!2!2!2]

Consider the String "COMPETITIVE". How many distinct strings can be made by rearranging its letters?

11!/[2!2!2!]

How many number 1-300 are divisible by 3, 7, or 11?

144

How many bit strings of length 10 contain either five consecutive 00's or five consecutive 11's ?

222

For a kid's party, you can choose between three themes, four colors for balloons, and two flavors of cake. How many different parties can you plan?

24

If we are labeling chairs with an uppercase letter followed by a number (1-100), how many different chairs can we uniquely label?

2600

How many ways are there to write a password that has at least 7 and at most 10 characters with only lowercase letters?

26^7 + 26^8 + 26^9 + 26^10

A system encodes a character using 8 bits, but the first 4 bits are reserved for future use (currently always 0). How many different strings can be represented with 32 bits?

2^16

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). A) A) How many ways are there to award the prizes? B) B) How many ways are there to award the prizes if the person holding ticket 47 wins the grand prize? C) C) How many ways are there to award the prizes if the person holding ticket 47 wins one of the prizes? D) D) How many ways are there to award the prizes if the person holding ticket 47 does not win a prize? E) E) How many ways are there to award the prizes if the people holding tickets 19 and 47 both win prizes? F) F) How many ways are there to award the prizes if the people holding tickets 19, 47, 73, and 97 all win prizes? G) G) How many ways are there to award the prizes if none of the people holding tickets 19, 47, 73, and 97 wins a prize? H) H) How many ways are there to award the prizes if the grand prize winner is a person holding ticket 18, 47, 73, or 97? I) I) How many ways are there to award the prizes if the people holding tickets 19 and 47 win prizes, but the people holding tickets 73 and 97 do not win prizes?

A) P(100,4) B) P(99,3) C) 4*P(99,3) D) P(99,4) E) 4*3*98*97 F) P(4,4) G) P(96,4) H) 4*99*98*97 I) 4*3*96*95

Let d be a positive integer. Show that among any group of d+1 (not necessarily consecutive) integers, there are at least two with the same remainder, when they are divided by d. A) Number of pigeons = B) Number of Holes = C) Equation?

A)d+1 B)d C) (d+1)/d = 2

How many ways are there to award 1st, 2nd, 3rd, and 4th place to 24 people (10 female, 14 male) if there has to be at least one male and at least one female that get awards?

C(10,1)*C(14,3)*4! + C(10,2)*C(14,2)*4! + C(10,3)*C(14,1)*4!

Your roommate sends you out to pick up plants. You both agreed on getting 8 plants, but didn't agree on which ones. If the store has cacti, air plants, lithops, and lucky bamboos, how many combinations of plants could you come home with?

C(11,3)

Roll 3 six-sided dice. How many di↵erent rolls can we have?

C(8,3)

Every day for a year, Hank chooses six distinct positive integers between 1 and 2006, inclusive. Which number is closest to the fraction of days that some pair of these integers has a difference that is a multiple of 5? Explain your answer for full credit. （A）12 （B）35 （C）23 （D）45 （E）1

E(1)

Let L(p,d) = "Person p lives with person d." Example Ap Ed L(p,d) Nancy lives with someone. (use d)

Ed L(Nancy,d)

Chris lives with at least two people. (use d and f; d & f are diff. is given)

EdL (Chris,d)*Ef L(Chris,f)

Chris lives with exactly two people. (use h)

EdL (Chris,d)*Ef L(Chris,f) ^ -(Eh L(Chris,h))

Let L(p,d) = "Person p lives with person d." Example Ap Ed L(p,d) There is someone who lives with everyone. (use p and d)

EpAd L(p,d)

Let L(p,d) = "Person p lives with person d." Example Ap Ed L(p,d) Nancy lives with Andy.

L(Nancy,Andy)

In a race of 20 runners, how many di↵erent first, second, third place results are there?

P (20, 3) = (20! / (20-3)!)

P(x) = "x runs 1 hour every weekday." Ax -P(x) a) Everyone does not run every weekday b) There is a person who will not run every weekday c) Everyone walks 1 hour every weekday d) There are no weekdays a person will run e) No one runs 1 hour every weekday f) No one runs at all g) Not everyone runs 1hr every weekday

e) No one runs 1 hour every weekday

A) Everybody at a company is assigned a unique 6 digit ID. How many unique IDs exist? (NOTE: You should also count the ID consisting of all zeros) b) If only IDs divisible by 2 are allowed, how many unique IDs exist? c) If only IDs divisible by 5 are allowed, how many unique IDS exist? d) If only IDs divisible by 5 and 2 are allowed, how many unique IDs exist? e) If only IDs divisible by 5 or 2 are allowed, how many unique IDs exist?

a) 1000000 b) 500000 c) 200000 d) 100000 e) 600000

a) In all parts of the following question, assume that all phone numbers being considered are from Wilmington, which has an area code of 910:Phone numbers consist of 10 digits. How many phone numbers are possible in Wilmington? b) How many phone numbers are possible in Wilmington if the number 7 cannot be used? c) How many phone numbers are possible in Wilmington if they must be divisible by 2 or 5? d) How many phone numbers are possible in Wilmington if there are no repeated digits?

a) 10^7 b) 9^7 c) 6000000 d) 5040

a) A bookshelf contains 3 different mathematics textbooks, 4 different engineering textbooks, and 5 different history textbooks. How many different ways are there to arrange the books on a shelf? b) How many different ways are there to arrange them if each book is sorted by category (so all of the books of a type are in a row)? c) How many different ways are there to arrange them if only the history books must be next to each other? d) How many different ways are there to arrange the books if none of the history books may be adjacent to each other?

a) 12! b) 5!*4!*3!*3! c) 5!*8! d) 7!*6720

There are 20 mathematic majors and 144 computer science majors at a college. A) How many ways are there to pick 4 representatives, so that 3 is/are mathematics majors and the other 1 is/are computer science majors? b) How many ways are there to pick two representatives who are either a mathematic major or a computer science major?

a) 164160 [C(20,3)∗C(144,1)] b) 13366 [C(164,2)]

a) License plates consist of three uppercase letters followed by four numbers. How many possible license plates can be made? b) If each state has a distinguishable license plate design, then between plate design and license ID, how many unique license plates are possible? c) A partial identification on an NC license plate reveals that it contains only vowels(AEIOU) and digits can only be even. How many NC plates could meet this requirement?

a) 175760000 b) 8788000000 c) 5^7

a) A Cookout Tray consists of one main item, two different sides, and a drink. If Cookout offers 10 main items, 10 sides, and 5 drinks, how many unique Cookout Trays are possible? (See Reminder below) b) For a slight upcharge on fries (one of the sides), cheese and/or chili may be added to fries. How many ways can you order fries if this is the case? c) How many different sides are there total? d) How many Trays total are possible now? e) Two of the sides, chicken nuggets and corn dogs, allow you to take one of six sauces. If you MAY take a sauce with each corn dog or chicken nugget side, then how many total sides are available? f) How many different trays are possible now? g) All 10 of the main items on the tray menu (except quesadillas) can come with any combination of 13 condiments. Quesadillas only come in two varieties. How many different main items are there? h) How many different trays are possible now?

a) 2250 b) 4 c) 13 d) 3900 e) 25 f) 15000 g) 73730 h) 110595000

a) How many bit strings of length 7 are there? b) How many different bit strings are there of length 7 that start with 0110? c) How many different bit strings are there of length 7 that contain the string 0000?

a) 2^7 b) 2^3 c) 20

a) If a deli offers 7 fillings, 7 types of bread, and 7 condiments, and each sandwich consists of a filling, a type of bread, and a condiment, how many different sandwiches can the sandwich shop create? b) What does the number of unique sandwiches become if a sandwich can have any combination of condiments (including no condiments)?

a) 343 b) 6272

a) How many numbers exist from 1 to 100 that are divisible by either 2 or 5? b) How many numbers exist from 1 to 100 that are divisible by either 3 or 4? c) How many numbers exist from 1 to 100 that are divisible by either 2, 3, or 5? d) How many numbers exist from 1 to 100 that are divisible by either 2, 3, or 4? e) How many numbers exist from 1 to 100 that are divisible by either 2, 3, or 8?

a) 60 b) 50 c) 74 d) 67 e) 67

A particular brand of shirt comes in 21 colors, has a male version and a female version, and comes in 4 sizes for each sex. How many different types of this shirt are made? a) Female shirts? b) Male shirts? c) Total shirts?

a) 84 b) 84 c) 168

a) A production line needs to manufacture 9 unique objects; and the order in which they are completed is important. In how many different orders can the production line manufacture the objects? b) In how many different orders can the objects be manufactured if exactly one of the objects must be completed immediately before another?

a) 9! b) 8!

In how many ways can a photographer at a wedding arrange 6 people in a row from a group of 10 people, where the bride and the groom are among these 10 people, if A) The bride must be in the picture b) Both the bride and groom must be in the picture? List guests first, then bride/groom. c) Exactly one of the bride and the groom is in the picture. List guests first, then bride/groom.

a) 90720 [ C(9,5)* C(1,1) * P(6,6)] b) 50400 [ C(8,4)* C(2,2) * P(6,6)] c) 80640 [ C(8,5)* C(2,1) * P(6,6)]

a) How many ways can you generate a team of 5 from 10 players? b) How many ways can you split 10 players into two teams (Team A and Team B) of 5 players each?

a) C(10,5) b) C(10,5)

Five friends go to a soccer game together. They have five seats in a single row. How many different ways can they sit at the game? How many ways can they sit if A+B need to sit beside each other?

a) P(5,5) b) P(2,2) * P(4,4)

P(x) = "x runs 1 hour every weekday." -Ax -P(x) a) Not all people run 1 hour every weekday b) Someone runs 1 hour every weekday c) Not everyone runs every hour every weekday d) No one runs 1 hour every weekday e) All people do not run 1 hour every weekday f) Not all people run during the week

b) Someone runs 1 hour every weekday

P(x) = "x runs 1 hour every weekday." ExP(x) a) Only one person runs 1 hour every weekday b) There is a person who runs 1 hour every weekday c) There is a person who runs 1 time d) A person will only run one day for the week e) All people run 1 hour every weekday f) There is a person who walks 1 hour every weekday

b) There is a person who runs 1 hour every weekday

P(x) = "x runs 1 hour every weekday." -Ex -P(x) a) There is no one who runs every hour during the week b) There is no one who does not run 1 hour every weekday c) There is no one who will run 1 hour every weekday d) There is someone who runs 1 hour every weekday e) Not everyone does not run 1 hour every weekday f) Everyone does not run 1 hour every weekday

b) There is no one who does not run 1 hour every weekday

L(x) = "x is a lion" D(x) = "x is a deer" A(x,y) = "x attacks y" R(x,y) = "x runs from y" AxAy(L(x) * D(y) * R(y,x)) -> -A(x,y) a) Deer run from lions who do not attack them b) Lions run from Deer who do not attack them c) Deer who run from lions cannot be attacked by them d) If any deer runs from some lion, all lions won't attack it e) If all lions runs from all deer, all deer won't attack them. f) Lions who run from deer cannot be attacked by them

c) Deer who run from lions cannot be attacked by them

What keywords do you look for when doing Combinations?

choose, subset

P(x) = "x runs 1 hour every weekday." AxP(x) a) A person will not miss a weekday of running b) All people run every day of the week c) All people run every hour d) All people run for 1 hour every weekday e) All people run 7 hours a week f) A person runs for all hours during the week

d) All people run for 1 hour every weekday

L(x) = "x is a lion" D(x) = "x is a deer" A(x,y) = "x attacks y" R(x,y) = "x runs from y" ExAyD(x) * L(y) * R(x,y) a) Some lion attacks some deer b) All deer are attacked by the same lion c) All deer run from all lions d) There is at least one deer who runs from all lions e) There is at least one deer who runs from some lion f) Only one deer runs from all lions g) Some deer run from some lions h)All lions attack some deer

d) There is at least one deer who runs from all lions

P(x) = "x runs 1 hour every weekday." -Ex P(x) a) There is no one who doesn't run every weekday b) There is no one who walks 1 hour every weekday c) There is no one who runs at all d) There is a person who does not like to run 1 hour every weekday e) There is no one who runs 1 hour every weekday f) There is a person who runs for only 1 hour g) There is a person who doesn't run 1hr every weekday

e) There is no one who runs 1 hour every weekday

P(x) = "x runs 1 hour every weekday." -Ax P(x) a) Not all people do not run 1 hour every weekday b) Someone runs 1 hour every weekday c) All people don't run 1 hour every weekday d) Not everyone walks but runs every weekday e) Not everyone runs every hour every weekday f) Not all people run 1 hour every weekday

f) Not all people run 1 hour every weekday

P(x) = "x runs 1 hour every weekday." Ex -P(x) a) There is a person who does not run every weekday b) There is no one who runs for 1 hour every weekday c) There is a person who walks 1 hour every weekday d) There is no one that doesn't run 1hr for no weekdays e) There is a person who jogs every weekday f) There is a person that does not run for 1hr every weekday

f) There is a person that does not run for 1hr every weekday

L(x) = "x is a lion" D(x) = "x is a deer" A(x,y) = "x attacks y" R(x,y) = "x runs from y" AxEyL(x) -> (D(y) * A(x,y)) a) Every lion attacks the same deer b) Some deer runs from every lion c) All lions attack two deer d) For every lion, there is only one deer it attacks e) Every lion attacks all deer f) All deer runs from the same lion g) For every lion, there is some deer it attacks

g) For every lion, there is some deer it attacks

L(x) = "x is a lion" D(x) = "x is a deer" A(x,y) = "x attacks y" R(x,y) = "x runs from y" ExAy(L(x) * D(y) * -A(x,y)) a) All deer are attacked by some lion b) All deer run from some lion c) The same lion attacks every deer d) There is some lion who attacks every deer e) All deer attack some lion f) The same lion attacks some deer g) There is some lion who fails to attack any deer

g) There is some lion who fails to attack any deer

L(x) = "x is a lion" D(x) = "x is a deer" A(x,y) = "x attacks y" R(x,y) = "x runs from y" ExEyL(x) * D(y) * A(x,y) a)Every Lion attacks one deer b)Some deer attack all lions c)Every lion attacks all deer d)Some deer runs from all lions e) Exactly one lion attacks some deer f)Every lion attacks every deer g)Exactly one lion attacks exactly one deer h)Some lion attacks some deer

h) Some lion attacks some deer

What does P(n,k ) equal written out?

n! / (n-k)!

What is C(n,k) equal to written out?

n! / (n-k)!k!

What keywords do you look for when doing permutations?

order, arrange, unique

List all of the permutations of {1,2,3}{1,2,3} with no replacement and cardinality 3.IMPORTANT: Be sure to use {} to enclose each permutation, and seperate terms and permutations with a ,, (comma). E.g., {1,2},{2,1}

{1,2,3}, {1,3,2}, {2,1,3}, {2,3,1}, {3,1,2}, {3,2,1}