Question.3210 - General Expectations: (i). For full credit, your written assignments must be accompanied by a narrative explanation/rationale for the process that you used to solve each problem. How did you choose the steps? What is the logic behind the choices that you made? Explain why the problems were solved the way they were solved. Use complete sentences, good English, and proper mathematical notation. (ii). For full credit, your written assignments must include the statement of each problem so the reader knows what you are trying to demonstrate. In cases where the assignment refers you to a book problem, you must also copy the statement of the appropriate problem in the book. Section 6.1: 1. Among 160 math students 45 have taken Discrete Math, 30 have taken History of Math. There are 12 have taken both. (a.) How many students have taken either Discrete Math or History of Math? (b.) How many students have taken only Discrete Math? (c.) How many students have taken only one of Discrete Math or History of Math? (d.) How many students have taken neither course? Section 6.2: 1. A pizza parlor offers whole–wheat, white and gluten–free crust options. The whole–wheat crust comes in three sizes, the white crust comes in four sizes and the gluten–free crust comes in two sizes. (a.) How many crust options does the parlor offer? (b.) Any pizza can be topped with exactly one of the following cheeses: full–fat mozzerella, reduced–fat mozzerella or soy cheese. How many choices does a customer have for a plain cheese pizza? (c.) Suppose that the pizza parlor offers 5 meat toppings and 7 vegetable toppings. How many ways can the customer choose a pizza with 3 distinct toppings? (Remember to factor in the choice of crust and cheese). (d.) How many ways can a customer choose a vegetarian pizza with 3 distinct toppings? (Remember to factor in the choice of crust and cheese). 2. How many numbers in the range 1000–9999 (a.) do not contain the same digit 4 times? (b.) end with an odd digit? (c.) end with an odd digit and have no repeated digits? (d.) have exactly three digits that are 7s? 3. Suppose that a password for a computer system must have exactly 8 characters. (a.) How many passwords are possible that only use lower–case alphabetic characters? (b.) How many passwords are possible that can use either upper– and lower–case characters? (c.) How many passwords are possible that use a combination of upper– and lower–case characters as well as exactly one numeric digit in the last position? (d.) How many passwords are possible that use a combination of upper– and lower–case characters as well as exactly one numeric digit if the position of the numeric digit is variable?1Section 6.3: Practice: Do #1 of Section 6.3 in the textbook and check your answer in the back of the book. 1. In a classroom of 30 students, show that there are at least two students with the same last initial. 2. One thousand students are transported on 15 busses. Show that there is a bus with at least 67 students. Each bus can hold a maximum of 80 students. Section 7.1: 1. Seven runners are in a race in which first, second and third place will be awarded. Assuming that there are no ties, how many different outcomes are possible? 2. Forty photographs are entered in a photo competition. How many ways are there to award the top eight places? 3. Three math students and two computer science students come to class late. (a.) There are five students in total. How many ways can the five students walk in through the door late (i.e. 1st, 2nd, 3rd, etc)? (b.) How many ways can the students come in, if a computer science student must be first? Section 7.2: 1. How many subsets with exactly 3 elements does a set with 10 elements have? 2. How many subsets with an odd number of elements does a set with 10 elements have? 3. How many subsets with at most 3 elements does a set with 100 elements have? 4. Do #9 of Section 7.2 in the textbook. Section 7.3: 1. Suppose a die with six sides (1,2,3,4,5,6) is tossed. (a.) What is the probability that the number that appears is a 3? (b.) What is the probability that the number that appears is divisible by 2? (c.) If the die is tossed twice, find the probability that the sum of the numbers appearing is at least 9. 2. Suppose a fair coin is tossed 4 times. (a.) Write out all possible outcomes (i.e. HHTT, etc). (b.) What is the probability of obtaining four heads? (c.) What is the probability of obtaining two heads and two tails? (d.) What is the probability of no heads? (e.) What is the probability of obtaining at least 1 head? 3. A bag contains 3 blue marbles, 2 green marbles, 1 white marble and 6 red marbles. (a.) Suppose a marble is selected from the bag and replaced. What is the probability of receiving a blue marble on the first selection and a blue marble on the second selection? (b.) Suppose a marble is selected from the bag and replaced. What is the probability of receiving a blue marble on the first selection and not a blue marble on the second selection? (c.) Suppose a marble is selected from the bag and replaced. What is the probability of receiving at least one blue marble and a white marble? (d.) Suppose a marble is selected from the bag and not replaced. What is the probability of receiving a blue marble on the first selection and a blue marble on the second selection?2
Answer Below:
Total xxxxxxxx Students xxxx have xxxxx maths x m xxxxxxxx that xxxx taken xxxxxxx of xxxxx P x Students xxxx have xxxxx both x m x Students xxxx have xxxxx only xxxxx - xxxxxxxx that xxxx taken xxxx history xx maths x a xxxxxxxx that xxxx taken xxxxxx math xx history x mUh x m x h xx m x - x Students xxxx have xxxxx only xxxxx - x Students xxxx have xxxx one xx discreet xxxx or xxxxxxx of xxxx P xxxx discrete xxxx P xxxx history xx math x Students xxxx neither xx course x mUh x a xxxxx of xxxxx option xxxxx wheat xxxxx and xxxxxx free x e x number xx of xxxxxxxxx sizes xxx whole xxxxx no xx different xxxxx for xxxxx crust xx of xxxxxxxxx sizes xxx glunten xxxxx therefore xx of xxxxxxxxx crust xxxxxx taking xxxx into xxxxxxxxxxxxx b xxxxxxxxx size xxx whole xxxxx Available xxxx for xxxxx crust xxxxxxxxx size xxx gluten xxxx crust xxx different xxxxx of xxxxxx toppings xx of xxxx of xxxxxxxxx a xxxxx from x Now xxx each xx choices xx have xxxxx of xxxxx toppings xxxxxxxxx no xx different xxxxxx customer xxx c xxxx topping xxx topping xxxxx no xx topping xxx Choosing xxxxxxxx topping xxxx can xx done xx C xxxx C x n xx of xxxx of xxxxxxxx n xxxxxxxx item xxxx m xxxxxxxxx itemsd xx of xxx topping xxx no xx ways xx choosing xxxxxxxx veg xxxxxxxx from xx C xxxx a xxxxx numbers xx the xxxxx to xxxxxxxxx both xxxxxxx containing xxx the xxxx digits xxxx So xx of xxxxxx not xxxxxxxxxx all xxxxxx similar x b xxxxx possible xxxxxxxxxxx of xxxxx leading xxxxxx in xxxx range xxx end xxxx or xxxx possible xxxx final xxxxxx or xx five xxxxxxxx odd xxxxx digits xx there xxx just xx many xxxxxx with xxx digits xx with xxxx ones xxxx exactly xxxx the xxxxxxx in xxx range xxx with xx odd xxxxx C xxxxx are xxxxxxxx odd xxxxx digits xxx each xx those xxxxx are xxxxxxxx first xxxxxx the xxxxx four xxx numbers xxxx and xxx each xxxxxxxx combination xx last xxx first xxxxx there xxx possible xxxxxxx for xxxx of xxx middle xxxxxx But xx can x choose xxx same xxxxx for xxxx so xxxxx are xxxx choices xxx the xxxx of xxxxxx digits xxxx there xxx numbers xx this xxxxx that xxx with xx odd xxxxx and xxxx no xxxxxxxx digits x There xxx possible xxxxx positions xxx the xxxx digit xx the xxxxx digit xxx t x there xxx only xxxxx possibilities xxx it xxxxxxx it xxx t xx zero xx If xxx non- xxxxx isn x first xxxxxxxxxxxxx there xxx other xxxxxxxxxxxxx for xx So xxxxx are xxxxxxx in xxxx range xxxx exactly xxxxxx that xxx s x no xx letters xx English xxxxxxxx s xxxx of xxxxxxxx Each xxxxxx in xxx password xxx be xxxxxxxx in xxxx Therefore xx of xxxxxxxxx possible xxxxxxxxx b xxxxxx both xxx cases xxxx consideration xxxxxx space xxxxxxx i x lower xxxx and xxxxx case xxxxxxxxx so xx of xxxx of xxxxxxxxx each xxxxxx in xxxxxxxx since xxxxxxxx has xxxxxxx total xxxxxxxx passwords x t xxx characters xx password xxxxx cahracters xxx alphabetic xx of xxxx of xxxxxxxx them xxx last xxxxxxxxx is xxxxx NO xx digits xxxxxxxxx possible xxxxxxxxx that xxx a xxxxxxxxxxx of xxxxxx and xxxxxxxxxx characters xx well xx exactly xxx numeric xxxxx in xxx last xxxxxxxx d xxxxxxx digit xxx occupy xx spaces x e xx ways xx Possible xxxxxxxxx that xxx a xxxxxxxxxxx of xxxxxx and xxxxxxxxxx characters xx well xx exactly xxx numeric xxxxx if xxx position xx the xxxxxxx digit xx variable x The xxxxx is xxxx simple xxxxx we xxxx only xxxxxxx in xxxxxxx alphabets xxx there xxx students xx atleast xxxxxxxx share xxxx last xxxxxxxx b xxxxxxxxx to xxx question xx the xxx transports xxxxxxxx Max xxxxxx of xxxxxxx that xxx be xxxxxxx in xxx with x constraint xx students xx there xx are xxxxx with xx more xxxxxxxx To xxxxxxx k xxxxxxxx objects xxxx n xxxxxxxx objects xxx formula xx given xx nPk xx k xx n xxxxx nPk x n-k xxx where x n xx n- x x x So xxx arranging xxxxxxxxx objects xxxx list xx is xxxxx by x - xxxxx reduces xx x x ways xxxx question xx similar xx and xxxx also xx use xxx permutation xxxxxxx for xxxxxxxxx things xxx of xxx answer xx given xx - x x xx significant xxxxxxx a xxxxx math xxxxxxx and xxx computer xxxxxxxx No xxxxxxxxx is xxxxx in xxx question xx each xx the xxx can xxxxxx any xx positions xxx boy xxx occupy xxx of xxxxxxxx So xxxx boy xxx occupy xxx remaining xxxxxxxx Similarly xxx third xxx can xxxxxx the xxxxxxxxx position xxx so xx Therefore xxxx and xxxx student xxx enter xxx class xx ways x for xx position xx need x computer xxxxxxx as xxx the xxxxxxxx hence xx position xxx be xxxxxxxx by xxxxxxxx out xx computer xxxxxxxx that xxxxxx late xxxx can xx done xx C xxxxxxx the xxxxxxxxx position xxx be xxxxxxxx in xxxx ways xx of xxxx in xxxxx student xxx enter xx computer xxxxxxx has xx st xxxx Choosing xxxxxxxx from x set xx can xx done xx C xxxx subset xxxx Odd xx elements xxx if xxx of xxxxxxx with xxxxxxxx Now xx of xxxx of xxxxxxxx - xxxxxxx C xxxxxxx C xxxxxxxx C xxxxxxxx C xxxxxxxx C xxxxx Now xxxxx ways xx subset xxxx odd xxxxx waysC xx most xxxxxxx means xxxx that xxx can xxxxxxx elements xxx a xxxxxx with xxxxxxx possible xx of xxx For x subset xxxx elements xxxxxxxx no xx elements x e xx element xxx be xxxxxx in xxxx nd xxxxxxx can xx chosen xx ways xxx they xxx be xxxxxxxx in xxxx For x subset xxxx elements xx of xxxxxxxx subsets xx total xxxx of xxxxxxx subset xxxx atmost xxxxxxx Sample xxxxx for xxxxxxxx dice xxxxxxxx a xxxxxxxxx of xx event xx given xx possible xxxxxxx divided xx the xxxxxx space x x xxxxxxxx outcome xxxxxx spaceProbablity xxxx the xx that xxxxxxx when x dice xx rolled x are xx i x divisible xx i x cases xxx of xx the xxxxxxxxx of x no xxxx appears x dice xxx is xxxxxxxxx by x Possible xxxx that xxx sum xx no xx greater xxxx or xxxxx to xx dice xx tossed xxxxx cases xxx sum xxxxx ways xxx equal xxxx sum xxxxx ways xxx equal xxx Total xxxxxxxx case xxx a xxxxx of xxxx and xxx atleast xxxx Sample xxxxx for xxx throw xx two xxxxx as xxxx the xxxx can xxxx possible xxxxxxxx respectively xxxxxxxxx probability xxx dices xxxxxx sum xx atleast x total xxxxxxxxx in xxxx No xx members xx committee xxx no xx ways xx choosing xxxxxxx from x a xxx a xxxx coin xxxxxxxx outcome xxxx or xxxx Probability xx getting x head xxxxxxxxxx of xxxxxxx a xxxx All xxxxxxxx outcomes xxx tossing xxx coin xxxx are xxxx HHHT xxxx HTHH xxxx HHTT xxxx HTTH xxxx THHT xxxx HTTT xxxx TTHT xxxx TTTTSo xx have xxxxx of xxxxxxxx outcomesb xxxxxxxxxxx of xxxxxxxxx all xxxx head xxxx ques x c xxxxxxxxxxx of xxxxxxxxx two xxxx and xxx tail xxxx rd xxxx of xxxx a x Probability xx no xxxx Probability xx all xxxx from xxxx ques x e xxxxxxxxxxx of xxxxxxx at xxxxx head xxxxxxxxxxx of xxxxxxx head xxxxxxxx ways xx getting xxxx ways xxxx ways xxxx ways xxxx way xx probability xx getting xx least xxxx a xxx bag xxxxxxxx blue xxxxx white xxx red xxxxxxx Total xxxxxxx marbles xx of xxxx of xxxxxxx blue xxxxxx Therefore xxxxxxxxxxx f xxxxxxx blue xxxxxx in xx attempt xxx after x draw xx replacement xx done xxxxxx space xxxxxxx the xxxx so xxxxxxxx of xxxxxxx blue xxxx in xxxxxx draw xx also xx combined xxxxxxxxxxx of xxxx marble xx st xxx nd xxxxxxx b xxxxx marbles xxxxxxx No xx ways xx getting xxxx marble xxxxxxxxx probability x getting xxxx marble xx st xxxxxxx Replacement xx done xx the xxxxxx space xxxxxxx the xxxx No xx ways xx selecting x non xxxx marble x So xxxxxxxxxxx of xxxxxxxxx a xxx blue xxxxxx is xxx using xxxxxxxx probability xxx a xxxx ball xx st xxxxxxx and x non xxxx ball xx second xxxxxxx is xxxxx by x question xxx clear xx no xx selections xxx mentioned x Total xxxxxxx marbles xx of xxxx of xxxxxxx blue xxxxxx Therefore xxxxxxxxxxx f xxxxxxx blue xxxxxx in xx attempt xxx after x draw xx replacement xx not xxxx So xx of xxxx balls xxxxxx Sample xxxxx becomes xxx the xxxxxxxxxxx of xxxxxxx blue xxxx in xxxxxx draw xx combined xxxxxxxxxxx of xxxx marble xx st xxx nd xxxxxxx when xxxxxxxxxxx is xxx done xxxxx newcomers xx club xx of xxxxxxx in xxxxxxxxx Now xx of xxxx of xxxxxxxx members xxxx C xxx problem xx a xxxx no xx days xx in x group xx members xxx people xxx have xxxxxxxxx days xx birth xx the xx person xxxxxxxxxxx shares xxx birthday xxxx some xxxxx personMore Articles From Maths
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