Name: Version #1. Please do NOT write in this box. Multiple Choice Total

Transcription

1 Name: Version #1 Instructor: Annette McP Math Exam 1 Sept. 13, The Honor Code is in e ect for this examination. All work is to be your own. Please turn o all cellphones and electronic devices. Calculators are allowed. The exam lasts for 1 hour and 15 minutes. Be sure that your name and your instructor s name are on the front page of your exam. Be sure that you have all 11 pages of the test. PLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 1. ( ) (b) (c) (d) (e) 2 ( ) (b) (c) (d) (e) 3. ( ) (b) (c) (d) (e) 4. ( ) (b) (c) (d) (e) 5. ( ) (b) (c) (d) (e) 6. ( ) (b) (c) (d) (e) 7. ( ) (b) (c) (d) (e) 8. ( ) (b) (c) (d) (e) 9. ( ) (b) (c) (d) (e) 10. ( ) (b) (c) (d) (e) Please do NOT write in this box. Multiple Choice Total

2 Name: Instructor: Math Exam 1 Sept. 13, The Honor Code is in e ect for this examination. All work is to be your own. Please turn o all cellphones and electronic devices. Calculators are allowed. The exam lasts for 1 hour and 15 minutes. Be sure that your name and your instructor s name are on the front page of your exam. Be sure that you have all 11 pages of the test. PLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 1. (a) (b) (c) (d) (e) 2 (a) (b) (c) (d) (e) 3. (a) (b) (c) (d) (e) 4. (a) (b) (c) (d) (e) 5. (a) (b) (c) (d) (e) 6. (a) (b) (c) (d) (e) 7. (a) (b) (c) (d) (e) 8. (a) (b) (c) (d) (e) 9. (a) (b) (c) (d) (e) 10. (a) (b) (c) (d) (e) Please do NOT write in this box. Multiple Choice Total

3 2. Initials: Multiple Choice 1.(5pts) Let U = {a, b, c, d,..., z} = The English Alphabet A = {m, a, t, h}, B= {i, s}, C= {l, i, t}, where U is the universal set. Consider the following statements: (1) B \ C is a subset of A. (2) (A \ B \ C) c = U. (3) A and B [ C are disjoint. (Note: (A \ B \ C) c =(A \ B \ C) 0 ). Which of the following is correct? (a) Only (2) is true. x (b) (2) and (3) are true and (1) is false. (c) (1) and (3) are true and (2) is false. (d) Only (3) is true. (e) All of the statements are true. Bric Ei This is not a subet of A Therefore statementfalse An Bnc to Thefore statementistree An Buc It f f Therefore statement e 2.(5pts) Of the 100 students in a dorm, 52 are currently enrolled in a math class, 30 are currently enrolled in a chemistry class and 25 are currently enrolled in both a math class and a chemistry class. How many students in the dorm are currently enrolled in neither a chemistry class nor a math class? (a) 43 (b) 57 (c) 7 (d) 75 (e) 0 M Students enrolled in MATH CLASS C a a a cheat CLASS We have n U too n M 52 M c 3 O and n Mrc 25 Question what is n Mvc

4 nlm c n M tn c n Mhc 52 t Therefore by The complement rule n u n Muc nkmuc

5 3. Initials: 3.(5pts) In the Venn Diagram below; the set R represents students at ND who play rugby, the set S represents students at ND who play squash, and the set T represents students at ND who play tennis. Which of the following sets is represented by the shaded region in the Venn diagram below? R S T (The word Students in the answers below should be interpreted as Students at ND.) (a) Students who play rugby and also play either squash or tennis (or both). (b) Students who play either squash or tennis (or both), but do not play rugby. (c) Students who play rugby, but do not play either squash or tennis. (d) Students who play all three sports. (e) Students who do not play any of the three sports. The shaded Region is in R and also in SVT fines or in Torino both It is infattall of RA SVT Thus it represents students who play Rugby and play either squash or Tennis or both 4.(5pts) There are 3 foil competitors, 3 sabre competitors and 2 epee competitors on a fencing team. The coach wishes to take a photograph of the competitors by lining them up with the foil competitors on the left, the sabre competitors in the middle and the epee competitors on the right. How many di erent such photographs are possibe? (a) 72 (b) 6 (c) 432 (d) 14 (e) 8! Foil Sabre epie stepl Line up the foil competitors 3 ways

6 step 2 Line up the Sabre competitors 3 ways step 3 Line up the epee competitors 2 ways Must complete ace 3 steps to take the photo Therefore by mutt principle Ther are 3 3 o pho

7 4. Initials: 5.(5pts) To place an order at Fred s Frozen Yoghurt bar, you must follow the rules below. First you must choose one container from 3 types of containers, cone, wa e cone or cup. Next; there are 10 di erent flavors of frozen yoghurt and you must choose to have at least one, but not more than 3 scoops of frozen yoghurt in your order. You may choose scoops of the same flavor or scoops of di erent flavors (e.g. you may go with 3 vanilla, or two vanilla and one chocolate, or one vanilla, one pistachio and one chocolate, or just two scoops of vanilla). Finally; there are 5 di erent types of toppings available and you may choose any subset of the 5 toppings (including no toppings ). How many di erent orders can you make? (a) 3 ( ) 2 5 (b) 3 (10 3 ) 2 5 (c) 3 (C(10, 1) + C(10, 2) + C(10, 3)) 2 5 (d) ! (e) 3 (P (10, 1) + P (10, 2) + P (10, 3)) 5! You must complete all of The Following steps to an place order step 1 Step 2 steps choose container choose 1,2 or 3 Scoops choose a repeats allowed subet of toppings 3 co ways flavours 5 toppings 6.(5pts) How many di erent words, including nonsense words can be made by rearranging the letters of the word WATTAMOLLA (a) 10! 2! 2! 3! (b) 10! (c) C(10, 2) C(10, 2) C(10, 3) (d) P (10, 2) P (10, 2) P (10, 3) (e) 2! 2! 3! 10 t ways 25 ways 1 Mutt principle 3 lt orders Different REARRANGEMENTS i w AAA USE ALL LETTERS L EEE

8 5. Initials: 7.(5pts) You have locked your valuables in a locker at the Smith Center and forgotten the 4-digit combination used. You remember that no digits were repeated in the combination, no the digit 0 was not used, repeats the combination either started with a 5 or ended with a 5 and all other digits were even. Either 5 How many 4-digit combinations fit your description? (a) 48 (b) 24 (c) (d) 4 3 (e) (5pts) Recall that a standard deck of 52 cards have 4 suits, Hearts, Diamonds, Clubs and Spades. Each suit has 13 cards. How many poker hands (5 cards drawn from a standard deck of 52) have 2 cards from one suit and three from another (for example 2 hearts and 3 spades). (a) 4 C(13, 2) 3 C(13, 3). (b) 4 P (13, 2) 3 P (13, 3). (c) 4 C(13, 2) + 3 C(13, 3). (d) C(13, 2) C(13, 3). (e) 4 P (13, 2) + 3 P (13, 3). Use only digits 1,2 3,4 5,478,9 3 digs from 12,4 6,8µg I To fill The Remaining 3 digits in a combo of the form 5 with no repeats using Elements of the set 2,4 6,83 we have 4 a ways we have Similarly 24 to complete ways a combination the I of form The Resulting sets of combos of the form Thus we have 5 are Disjoint IN possible combinations Principle Malt Step Pick a suit ways Step 2 choose 2 cards CCB 2 ways Step 3 Pick A Different suit 3 ways 5 and step choose 3 cards from second suit 2413,3 ways ALL It cc 13,2 3 C 13,3 ways

9 1 6. Initials: 9.(5pts) An experiment consists of flipping a coin 12 times and observing the sequence of heads and tails. How many such sequences have at least 3 heads? (a) (b) (c) 79 (d) C(12, 3) (e) P (12, 3) At least 3h's Efton Ett's sequences of this type t CCR l t 12,27 c 12,0 I tiz t Thus with at Least 3 H's sequences 2 79 Tot sequences 3 10.(5pts) A quality control inspector takes a sample of 10 widgets for testing, from a box containing 50 widgets. How many di erent samples are possible? (a) C(50, 10) (b) P (50, 10) (c) 50! (d) (e) 500 There are CC50,10 ways to choose a io sample of objects from a set of 50

10 7. Initials: Partial Credit 11.(12pts) A survey of 100 faculty, on whether they bought lunch at the on-campus dining facilities named below, in the past week, revealed the following; 30 of them bought lunch in the Duncan Center, 25 of them bought lunch in the Huddle, and 22 of them bought lunch at Grace Hall. Seven of those interviewed bought lunch at Grace Hall and the Duncan Center, 9 of them bought lunch at the Duncan Center and at the Huddle, and 10 of the bought lunch at the Huddle and at Grace Hall. Three of the faculty interviewed bought lunch at all three facilities n the past week. (a) Represent this in the diagram shown on the left below. t ANGAD 3 U U G D 100 H H G µ 3 b I 46 g I I D G = Grace Hall H = Huddle D = Duncan Center G = Grace Hall H = Huddle D = Duncan Center (b) Shade the region in the diagram on the right above which corresponds to faculty who bought lunch at exactly one of the 3 dining facilities in the past week. (c) How many of the faculty interviewed did not buy lunch at any of the dining facilities named above in the past week? 46

11 8. Initials: 12.(12pts) Alan is going to go for a jog through his neighborhood. Alan always jogs efficiently and chooses a route that does not backtrack (that is, he only ever jogs South or East). A E S B C D The questions below refer to the street map shown above. Your answers may be given in terms of numbers and symbols of the form P (n, r), C(n, r) and factorials. (a) How many jogging routes can Alan take from his home A to that of his friend Daniel at D? 13 blocks Routes from A tod A D is the number ofways to travel which are southcs of which are East E AND 5 of 5 1,287 Thus Routes A D 13,8 CCI3 8 5 (b) How many routes can Alan take if he intends to jog from his home A to that of his friend Daniel at D, and he intends to stop briefly at Ben s home at B. ALAN Must To Jog from A D Thru B Always going SORE B and then go from A multiplication A principle D thru ways B from we get CALIg CB 5 first Using the D B (c) How many routes can Alan take if he intends jog from his home A to his friend Daniel s house at D, stopping briefly at Ben s home at B, but avoiding the intersection at C. D as before Except ALAN must JOG FROM A B AND THEN from B at C on THE second INTERSECTION This Time He Must Avoid THE of his tourney Leg the the minus HE ta ofr B D of feasibleroutes from C THE he must Avoid is 8,5 eg B THE TOTAL 24 4,3 c c 4,21 24 D of such routes can take Thus THE A Routes from A D THAT ALAN J C 5,3 CC8,5 Caio CC4,2 is A B

12 9. Initials: 13.(12pts) How many di erent words with three letters, including nonsense words, can be made from the letters of the word PHENOMENAL if: (a) Letters CANNOT be repeated. 11 PHENOM AL 8 Letters 8 I I P sis words (b) Letters CANNOT be repeated and the word must start with an E. E I (c) Letters CANNOT be repeated and the middle letter must be an E. I E 6 442J (d) Letters CANNOT be repeated and at least one letter must be an E. These Are 3 Letter words with Exactly owe E since Letters cannot be repeated The E is either The 1st Letter 62 such words a 2W a 142 n 3rd n 42 Disjoint words with at Least one E J

13 10. Initials: 14.(12pts) A bag contains 10 red marbles (numbered 1-10) and 5 green marbles (numbered 1-5). (a) How many di erent samples of 5 marbles can be drawn from the bag? 5 Sample size to Esainalesmoftasizeafagthaottease CCI (b) How many di erent samples of 5 marbles, with 2 red marbles and 3 green marbles, can be drawn from the bag? in 2 steps we can create a sample with LR and 3G 2 IOR's 45 ways Clio 212 from the step1 Draw to ways Ccs 3 5 G's the 2 Draw 3G from step By The multi principle we have Such samples TT 450 ((b) How many di erent samples of 5 marbles, with at least 2 red marbles, can be drawn from the bag? Samples with 212 Exactly at least 212 marbles can have marbles or Exactly 312 or Exactly 4R or Exactly 5R It is easier to count the number The complement of this set is of samples in the set of the complement here sampleswit either with hessthawzr.ie such sample 5 Green marbles CC5,51 1 OR marbles Exactly 50 such samples 41 CK marbles l and Clio 4G IR IR marble or Exactly samples Thus and the the of samples in the complement 51 at Least 212 with samples of 51 samples of sizes Total 425,5 51 3,008 12,

14 11. Initials: 15.(2pts) You will be awarded these two points if you write your name in CAPITALS and you mark your answers on the front page with an X (not an O). You may also use this page for ROUGH WORK