MPC 1 PERMUTATION AND COMBINATION Syllabus : Fudametal priciples of coutig; Permutatio as a arragemet ad combiatio as selectio, Meaig of P(, r) ad C(, r). Simple applicatios.
Permutatios are arragemets ad combiatios are selectios. C1 Fudametal Priciple ad Coutig : (i) Priciple of Multiplicatio : MPC C O N C E P T S If a evet ca occur i m differet ways, followig which aother evet ca occur i differet ways, the total umber of differet ways of simultaeous occurrece of both the evets i a defiite order m. (ii) Priciple of Additio : C Arragemet : If a evet ca occur i m differet ways, ad aother evet ca occur i differet ways, the exactly oe of the evets ca happe i m + ways. If P r deotes the umber of permutatios of differet thigs, takig r at a time, the P r C3 Circular Permutatio :! ( 1)( )...( r 1) ( r)! The umber of circular permutatios of differet thigs take all at a time ( 1)! If clockwe ad ati-clockwe circular permutatios are cosidered to be same, the it ( 1)! Number of circular permutatio of thigs whe p alike ad the rest differet take all at a time dtiguhig clockwe ad aticlockwe arragemet C Selectio : ( 1)! p! If deotes the umber of combiatios of differet thigs take r at a time, the C! r!( r)! P r! r r where r The umber of permutatios of thigs, take all at a time, whe p of them are similar ad of oe type, q of them are similar ad of aother type, r of them are similar ad of a third type ad the remaiig (p + q + r) are all differet C5 Formatio of Groups :! p!q!r!. Number of ways i which (m + + p) differet thigs ca be divided ito three differet groups cotaiig m, ad p thigs respectively ( m p)!, m!!p!. If m = = p ad the groups have idetical qualitative charactertic the the umber of groups = (3)!.!!!3! (3)! However, if 3 thigs are to be divided equally amog three people the the umber of ways = 3 (!)
C6 Selectios of oe or more objects Number of ways i which atleast oe object be selected out of dtict objects C 1 + C + +... + C = 1 MPC 3 Number of ways i which atleast oe object may be selected out of p alike objects of oe type q alike objects of secod type ad r alike of third type (p + 1) (q + 1) (r + 1) 1 Number of ways i which atleast oe object may be selected from objects where p alike of oe type q alike of secod type ad r alike of third type ad rest (p + q + r) are differet, (p + 1) (q + 1) (r + 1) (p + q + r) 1 INITIAL STEP EXERCISE 1. The umber of ways i which cadidates A 1, A,...,A 10 ca be raked if A 1 always above A 5. 8! 5.! 5 5 + 8!. The umber of ways i which 9 differet pearls ca be joied to form a ecklace, so that particular pearls caot be seperated 5! 5! 10 9!/! 880 3. The umber of eve atural umbers havig three digits if repetitio of digits allowed 50 350 550 oe of these. There are four orages, five apples ad six magoes i a fruit basket. The umber of ways ca a perso make a selectio of fruits amog the fruits i the basket? 10 90 09 90 5. The adjacet figure to be coloured usig three differet colours. The umber of ways i which th ca be doe if o two adjacet triagles have the same colours 7. There were two wome participatig i a chess touramet. Every participat played two games with the other participats. The umber of games that the me played betwee themselves proved to exceed by 66 the umber of games that the me played with the wome. The umber of participats 6 11 13 oe of these 8. The total umber of seve digit umbers the sum of whose digits eve 9 10 6 5 10 5 81 10 5 9 10 5 9. P a set cotaiig elemets. A subset A of P choose ad the set P costructed by replacig the elemets of A. A subset B of P choose. The umber of ways of choosig A ad B such that A ad B have o commo elemet 3 oe of these 10. The umber of differet ways the letters of the word VECTOR ca be placed i the 8 boxes of the give figure so that o row empty equal to 81 6 oe of these 6. A lady gives a dier party to 5 guests to be selected from ie frieds. The umber of ways of formig the party of 5, give that two of the frieds will ot atted the party together 56 91 16 oe of these 6 6! 5 6! 7 6! oe of these 11. Number of itegral solutios of equatio x + y + z + t = 0, where x, y, z, t are all 1, 7 7 C 1 7 C 7
1. Te differet letters of a alphabet are give. Words with five letters are formed from these give letters. The the umber of words which have at least oe letter repeated 69760 300 9978 oe of these 13. There are 10 lamps i a hall. Each oe of them ca be switched o idepedetly. The umber of ways i which the hall ca be illumiated 10 103 10 10! MPC 1. The total umber of 9 digit umbers which have all differet digits 10! 9! 9.9! 10.10! 15. A five-digit umber divible by 3 to be formed usig the umerals 0, 1,, 3, ad 5, without repetitio. The total umber of ways th ca be doe 16 0 600 315 FINAL STEP EXERCISE 1. The umber of eve divors of 1600 1 18 3 oe of these. Give that odd, the umber of ways i which three umbers i A.P. ca be selected from 1,, 3,... ( 1) ( 1) 3. The expoet of 3 i 100! 33 8 5 ( 1) ( 1). The umber of permutatios that ca be formed by arragig all the letters of the word NINETEEN i which o two E s occur together 8! 3!.3! 5! 8 C3 3! 5! 6 8! C3 6 C3 3! 5! 5. If oe quarter of all three elemet subsets of the set A = {a 1, a, a 3...,a } cota the elemet a 3, the = 10 1 1 oe of these 6. The umbers are picked at radom from the umbers 1,, 3,...,100. The umber of ways of selectig the two umbers whose product a multiple of 3 58 11 739 oe of the above 7. The umber of ways i which dtict objects ca be put ito two idetical boxes so that o box rema empty 1 1 1 8. By usig the digits 0, 1,, 3, ad 5 (repetitios ot allowed) ay umber of digits beig used, the umber of o-zero umbers that ca be formed 1030 1630 100 1530 9. The umber if triagles whose vertices are at the vertices of a octago but oe of whose sides happe to come from the sides of the octago 5 8 16 10. There are three piles of idetical red, blue ad gree balls ad each pile cota at least 10 balls. The umber of ways of selectig 10 balls if twice as may red balls as gree balls are to be selected 3 6 8 11. If objects are arraged i a row, the the umber of ways of selectig three of these objects so that o two of them are ext to each other 3 C 3 oe of these 1. The letters of the word SURITI are writte i all possible orders ad these words are writte out as i a dictioary. The the rak of the work SURITI 5 307 315
13. A cadidate required to aswer 6 out of 10 questios which are divided ito two groups each cotaiig 5 questios ad he ot permitted to attempt more tha from each group. The umber of ways he ca make up h choice 150 00 50 300 1. 0 persos where ivited for a party. The umber of ways i which they ad the host ca be seated at a circular table such that two particular persos be seated o either side of the host 0! 19! (18!) 18! 15. Give 5 differet gree dyes, four differet blue dyes ad three differet red dyes. The umber of combiatios of dyes which ca be chose takig at least oe gree ad oe blue dye 3600 370 3800 Noe MPC 5 16. The sides AB, BC, CA of a triagle ABC have 3, ad 5 iterior poits respectively o them. The total umber of triagles that ca be costructed by usig these poits as vertices 17. k 0 0 195 05 k m k C r = + 1 m m + 1 m ANSWERS (INITIAL STEP EXERCISE) 1. a. b 3. d. a 5. b 6. b 7. d 8. b 9. a 10. a 11. d 1. a 13. b 1. c 15. a ANSWERS (FINAL STEP EXERCISE) 1. b 6. c 11. a 16. c. d 7. c 1. a 17. d 3. c 8. b 13. b. d 9. d 1. c 5. d 10. b 15. b
MPC 6 AIEEE ANALYSIS [003] 1. The umber of ways i which 6 me ad 5 wome ca die at a roud table if o two wome are to sit together give by 30 5!! 7! 5! 6! 5!. A studet to aswer 10 out of 13 questios i a examiatio such that he must choose at least from the first five questios. The umber of choices available to him 196 80 36 10 3. If deotes the umber of combiatios of thigs take r at a time, the the expressio + 1 + equals + + 1 + 1 + AIEEE ANALYSIS [00/005]. How may ways are three to arrage the letters i the work GARDEN with the vowels i alphabetical order? 360 0 10 80 [00] 5. The umber of ways of dtributig 8 idetical balls i 3 dtict boxes so that oe of the boxes empty 3 8 1 5 8 [00] 6. The rage of the fuctio f(x) = 7 x P x 3 {1,, 3} {1,, 3,, 5, 6} {1,, 3, } {1,, 3,, 5} [00] 7. If the letters of the word SACHIN are arraged i all possible ways ad these words are writte out as i dictioary, the the word SACHIN appears at serial umber 600 601 60 603 [005] AIEEE ANALYSIS [006] 8. At a electio, a voter may vote for ay umber of cadidates, ot greater tha the umber to be elected. There are 10 cadidates ad are to be elected. If a voter votes for at least oe cadidate, the the umber of ways i which he ca vote 1110 500 610 385 9. The set S : = {1,, 3,..., 1} to be partitioed ito three sets A, B, C of equal size. Thus, A B C = S, A B = B C = A C =. The umber of ways to partitio S 1! 3 (!) AIEEE ANALYSIS [007] 1! 3 3!(!) 1! (3!) 1! 3!(3!) ANSWERS AIEEE ANALYSIS 1. d. a 3. a. a 5. b 6. a 7. b 8. d 9. a
MPC 7 TEST YOURSELF 1. The umber of diagoals of a polygo of 15 sides 105 90 75 60. The umber of 10 digit umbers that ca be writte by usig the digits 1 ad 10. The umber of six-digit umbers which have sum of their digits as a odd iteger, 5000 50000 97000 970000 10 C + 9 C 10 10 10! 3. The greatest umber of poits of itersectio of circles ad m straight lies m + m C 1 m(m 1) + (m + 1) m C + ( C ) oe of these. The umber of atural umbers with dtict digits 9 10 1 10 10 9 10 9 10 1 oe of these 5. The umber of arragemets of the letters of the word BANANA i which two N s do ot appear adjacetly 0 60 80 100 6. The umber of ways we ca put 5 differet balls i 5 differet boxes such that at most three boxes empty, equal to 5 5 5 5 5 10 5 oe of these 7. The total umber of ways of dividig m dtict objects ito equal group (m)! (m!) (!) (!) (m)! m (m!) (m)! m!! oe of these 8. The umber of ways i which three girls ad ie boys ca be seated i two cars, each havig umbered seats, 3 i the frot ad at the back, 1 P 1 1 P 1 1 3 11 P 9 oe of these 9. A library has differet books ad has p copies of each of the book. The umber of ways of selectig oe or books from the library p p ( + 1) p 1 (p + 1) 1 1. b. b 3. b. d 5. a ANSWERS 6. a 7. a 8. a 9. d 10. b