exam 2 solution - 22M2150:001 Fall 2012 Exam 2 Duration of...

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Unformatted text preview: 22M2150:001 Fall 2012: Exam 2 Duration of exam: 50 minutes 318t October, 2012 Name: 50LUTION$ Instructions: 1. There are a total of 6 questions and the maximum marks for the exam is 50. The maximum marks for each question is listed below the ques— tion; in order to get full credit, you must show detailed work unless otherwise stated. For questions 2 to 6, partial credit will be awarded provided there is enough evidence of clear, legible work leading to the solution. 2. You are NOT allowed to use books, class notes or any calculating aids. You are also not allowed to use mobile phones or any other electronic devices during the examination. 3. If you have any other doubts regarding the question paper, please feel free to ask me. 4. IMPORTANT: Use of unfair means will not be tolerated at any cost — any such incident will be treated as per the CLAS Code of Academic Honesty. ALL THE BEST! 1. Provide the answers to the following questions. Your answers should be in simplified form i.e., you should evaluate all sums7 products, fac- torials, binomial coeffecients, etc. (You do not need to show your work for this question.) (a) The number of 3-digit numbers is q 0 o (b) The number of ways one can choose a committee of 4 from a set of 7 people is 35 - (c) The number of ways one can seat 5 people around a round table is 214 (d) The number of factors of 420 (including 1 and 420) is Z Ll . (e) The number of nonempty subsets of a Set with 6 elements is 63 (f) The number of possible arrangements of (all) the letters of the word PEACE is 50 (g) The number of injective functions from a Set with 4 elements to a set with6elementsis 35° (2X7214 marks.) EAC' o) “I “1-1040 :l‘lool g) PE ‘1 no H) is 6+ 5‘ ‘7) l4 3 }/2 g M 3‘ s c). E}. W433 u 5‘ .3 ll ‘ é'g a) 42w 22 3"5 7 2. (a) What is the total number of ways to place 8 different books on 6 different shelves? (b) What are the total number of ways to travel in myzw—space from the point (1, 3, 2, 2) to the point (3, 7, 4, 3) by taking steps of one unit in the positive x—, y—, z- or w~directions? (Note: Moving in the negative x—,y—,z— or w—directions is prohibited.) (c) What is the total number of 7—digit palindromes with the restric- tion that no digit may appear more than twice. (01) Suppose there is a collection of 7 different French books, 5 differ— ent German books and 8 different Spanish books. How many ways can one choose two books such that the books are from different languages? Simplify your answer. (3 X 4 = 12 marks.) Tm. gv'rsl‘ (cook Can. be 19(0ch bu gum-ado] 44m. lasf (Ea/Mk) L“) in mt 1» 7" war, loook Cam 19¢ Flamed QM {3 “0‘30- l'otc-L 43? al- l 13 14' of. 0.10.»? 1L0 Place. ’l’he books ‘ 3 84'6") : d 8. ( 8 7 K it a" fi 0‘, wa-aa +0 WW pm hm books lo OF 1"“ “*5 +970 fixed Mamme (Wm/Mk3 i“ W). LC) (3—)) :2 5M»: w {W— Fosihw xgmwm (7‘3)31{ H Ii u l 7—dame (4'2)=2 u u " n z—owabm (34% WP ~ " ~ w—meav-m. th tra; wag/3 +0 Ham/{J : (“WI”)! 67! 2.14,;2312 42222! n 5 q (M) L,L.J./ (104)00—3} V7 S’MO— U” 1” a" Sin; [hr Fah'ndwme. M01134"? Maj “‘PPW Mare Jr“ km.“- 13 05— 3,1—0‘4‘3/(7L ?0bndmw M'H‘ “HM” MMJ‘M : ‘1' 8' 2} TM, hag loo<>K§ Um“ be FY + Gar Crew ’r SF Fv+ 5? , 0M5! SMCLHuL Wm doe: m9” mafi‘w/ rm” #0} “0‘30 5 Wm m (cooks 0: 15+ 98+7-82113L} 3. Show that in any set of n + 1 positive integers not exceeding 271, there must be two that are relatively prime. (Note: The number 1 is rel— atively prime to every positive integer because the greatest common divisor of 1 and any other positive integer is 1.) (5 marks.) ‘ 2 +02“ La's anth m 2" we?” gm W10 V‘ Serb no 30000351 CK,K-H):) if: MFG“ a) 2< 0m! AIKH The” di((i<+l)-4k) :9 all: 4. For What natural numbers n, is 2" > 71.2? Prove your result. 2‘71”/2.‘:z‘,23<3 ,1 ’ x ’ ’ (JmM W n 7.. >5. MA “:1) H'vaz we W 44W" 2 7” 7C” n/ ( we wcwdr Lo 9km») 2k“ >Q‘H) / L 2k+‘ ; 2 2 k >kL) 7 2K; (’ Z 7' k. 7 kLJrLIk C k’n/k >4) >9 +ZKH L 717‘) ;Q4+1)L \c‘ 5" 2 70:44) 5. In how many ways can one distribute 10 indistinguishable balls into 4 distinguishable boxes 31,32:ng and B; such that: (2X428marks.) 1”“ on}. (a) We we looking {OVA Soluh‘ans ,0 Jke Qfiucfiam X +X‘L +X3 +X9510 Xi}! L=i,7—,3,lf 3 W‘T H” 4w. Some 0/) ' r "r. M mm TM 'w #13 1%“- A )Guxwfmm) ’13 g .:x ~)) +)/ +Y3 +74‘ 6 Yc'7’o (YL L y' z i=‘/2,3I’§ Elli/3,5" ’ —l 0! “I 8’ 7 4m Wear 1; (6+4 ) I W :M WV- W5 “9}. TIM. W6} out»? In, Cash“: A 0k lug? W box 8,02,) I :‘ 701764, if 0;, odour k; dgwbbf; Mae \oaJS lo ~ 1‘40? cucuf «Lo ouchs‘lowfl Sud» Had? (we? 0X W ck 12.0% me balk 6. Prove the following identity: WG— l/JOM} h) dnooya O~ Cemmbfee. of. m 1925,9111. Rom 4km Th2- fi 0}— wwf: 1% do So u ("‘3") Ana» may we mm Auk oi 4M .Www 6r “L” Wm“ M O (cu-<3} mm 1 New n1 “We” 3' m w mm.) = m Z MGM "‘ 2' W“ I 2 o m WU) mm (,1 en n M“ E W MOA <5? coumHu/IJ/ 4‘“ it? Of wa‘ao n M_ a ‘ (“3) (»)*m('3)+ (M?) ...
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