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 xxx h xxxxxxxx that xxxx taken xxxx maths x Students xxxx have xxxxx only xxxxxxx of xxxxx - x Students xxxx have xxxxx either xxxx or xxxxxxx P xxx P x P x -P x cap 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 xxxxx P xxx - x Types xx crust xxxxxx Whole xxxxx white xxx gluten xxxx i x n xxxxxx no xx different xxxxx for xxxxx wheat xx of xxxxxxxxx sizes xxx white xxxxx no xx different xxxxx for xxxxxxx crust xxxxxxxxx no xx different xxxxx option xxxxxx size xxxx consideration x Available xxxx for xxxxx wheat xxxxxxxxx size xxx white xxxxx Available xxxx for xxxxxx free xxxxx Now xxxxxxxxx types xx cheese xxxxxxxx No xx ways xx selecting x crust xxxx a xxx for xxxx of xxxxxxx we xxxx types xx pizza xxxxxxxx Therefore xx of xxxxxxxxx choice xxxxxxxx has x meat xxxxxxx veg xxxxxxx Total xx of xxxxxxx Now xxxxxxxx distinct xxxxxxx from xxx be xxxx in x ways x m x ndash xx of xxxx of xxxxxxxx lsquo x rsquo xxxxxxxx item xxxx lsquo x rsquo 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 xxx t xxxxxx the xxxx digit xxx both xx there xxx only 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 amp x a xxxxx are xxxx other xxxxxxxxxxxxx for xx because xx can xxx t xx zero xx If xxx non- xxxxx isn xxx t xxxxx possibilities xxxxx are xxxxx possibilities xxx it xx there xxx numbers xx this xxxxx with xxxxxxx digits xxxx are x a xx of xxxxxxx in xxxxxxx alphabet xxxxx 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 xxxx c x tal xxxxxxxxxx in xxxxxxxx first xxxxxxxxxx are xxxxxxxxxx No xx ways xx choosing xxxx Now xxxx character xx digit xx of xxxxxx Therefore xxxxxxxx passwords xxxx use x combination xx upper- xxx lower-case xxxxxxxxxx as xxxx as xxxxxxx one xxxxxxx digit xx the xxxx position x numeric xxxxx can xxxxxx of xxxxxx i x in xxxx so xxxxxxxx passwords xxxx use x combination xx upper- xxx lower-case xxxxxxxxxx as xxxx as xxxxxxx one xxxxxxx digit xx the xxxxxxxx of xxx numeric xxxxx is xxxxxxxx a xxx logic xx very xxxxxx since xx have xxxx letters xx English xxxxxxxxx and xxxxx are xxxxxxxx so xxxxxxx students xxxxx same xxxx initials x according xx the xxxxxxxx if xxx bus xxxxxxxxxx students xxx number xx student xxxx can xx carried xx bus xxxx a xxxxxxxxxx of xxxxxxxx So xxxxx is xxx buses xxxx or xxxx students xx arrange x distinct xxxxxxx from x distinct xxxxxxx the xxxxxxx is xxxxx by xxx amp xx k xxx lt x where xxx n xxx and xxxxx n x n- xx x x x xx for xxxxxxxxx different xxxxxxx from xxxx of xx given xx P x which xxxxxxx to x x xxxx This xxxxxxxx is xxxxxxx to xxx here xxxx we xxx the xxxxxxxxxxx formula xxx permuting xxxxxx out xx the xxxxxx is xxxxx by x P x to xxxxxxxxxxx figures x Three xxxx student xxx two xxxxxxxx students xx condition xx given xx the xxxxxxxx so xxxx of xxx boy xxx occupy xxx of xxxxxxxxx One xxx can xxxxxx any xx position xx next xxx can xxxxxx the xxxxxxxxx position xxxxxxxxx the xxxxx boy xxx occupy xxx remaining xxxxxxxx and xx on xxxxxxxxx math xxx comp xxxxxxx can xxxxx the xxxxx in xxxx b xxx st xxxxxxxx we xxxx a xxxxxxxx student xx per xxx question xxxxx st xxxxxxxx can xx occupied xx choosing xxx of xxxxxxxx students xxxx turned xxxx This xxx be xxxx in x waysNow xxx remaining xxxxxxxx can xx occupied xx Ways xxxx No xx ways xx which xxxxxxx can xxxxx if xxxxxxxx student xxx to xx ways xxxxxxxx elements xxxx a xxx of xxx be xxxx in x ways xxxxxx with xxx no xxxxxxxx out xx set xx subsets xxxx elements xxx no xx ways xx choosing x element x element x elements x elements x elements x waays xxx total xxxx of xxxxxx with xxx terms xxxxx At xxxx element xxxxx that xxxx set xxx contain xxxxxxxx For x subset xxxx element xxxxxxxx no xx set xxx a xxxxxx with xxxxxxxx possible xx of xxxxxxxx i x st xxxxxxx can xx chosen xx ways xx element xxx be xxxxxx in xxxx and xxxx can xx arranged xx Ways xxx A xxxxxx with xxxxxxxx no xx possible xxxxxxx So xxxxx ways xx forming xxxxxx with xxxxxx element xxxxxx space xxx unbiased xxxx outcomes x probality xx an xxxxx is xxxxx by xxxxxxxx outcome xxxxxxx by xxx sample xxxxx P x Probable xxxxxxx sample xxxxxxxxxxxxxxx that xxx no xxxx appears xxxx a xxxx is xxxxxx b xxx no x e xxxxxxxxx by x e xxxxx out xx SO xxx probality xx a xx that xxxxxxx o xxxx and xx divisible xx frac 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 frac xxx possible xxxxxxxx for xxxxxxx the xxxx time xxx HHHH xxxx HHTH xxxx THHH xxxx HTHT xxxx THTH xxxx TTHH xxxx THTT xxxx TTTH xxxxxx we xxxx total xx possible xxxxxxxxx Probability xx obtaining xxx four xxxx from xxxx a x probability xx obtaining xxx head xxx two xxxx from xx case xx ques x d xxxxxxxxxxx of xx head xxxxxxxxxxx of xxx tail xxxx case xxxx A x possibility xx getting xx least xxxx Possibility xx getting xxxx Possible xxxx of xxxxxxx head xxxx head xxxx head xxxx head xxx So xxxxxxxxxxx of xxxxxxx at xxxxx head x the xxx contains xxxx green xxxxx and xxx marbles xxxxx marbles xxxxxxx No xx ways xx getting xxxx marble xxxxxxxxx probability x getting xxxx marble xx st xxxxxxx frac xxx after x draw xx replacement xx done xxxxxx space xxxxxxx the xxxx so xxxxxxxx of xxxxxxx blue xxxx in xxxxxx draw xx also xxxx So xxxxxxxx probability xx blue xxxxxx in xx and xx attempt xxxx frac x Total xxxxxxx marbles xx of xxxx of xxxxxxx blue xxxxxx Therefore xxxxxxxxxxx f xxxxxxx blue xxxxxx in xx attempt xxxx 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 xxxx Now xxxxx combined xxxxxxxxxxx for x blue xxxx in xx attempt xxx a xxx blue xxxx in xxxxxx attempt xx given xx frac xxxx c xxxxxxxx not xxxxx as xx of xxxxxxxxxx not xxxxxxxxx d xxxxx marbles xxxxxxx No xx ways xx getting xxxx marble xxxxxxxxx probability x getting xxxx marble xx st xxxxxxx frac 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 xxxx total xxxxxxxxx in xxxx No xx members xx committee xxx no xx ways xx choosing xxxxxxx from x Mod xxxxxxx In x week xx of xxxx So xx a xxxxx of xxxxxxx max xxxxxx can xxxx different xxxx of xxxxx So xxx th xxxxxx neccesarily xxxxxx its xxxxxxxx with xxxx other xxxxxxMore Articles From Maths
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