LASS: XI: MATHEMATIS ERMUTATIONS AND OMBINATIONS RATIE QUESTIONS ON FATORIAL AND FUNDAMENTAL RINILES OF OUNTING ove the followig fo N, 1. (2 )! 2.!.[1.3.5...(2 1)] FORMULA USED Factoial otatio:! o 1.! ( 1) 2)( 3)...3.2.1 2. 0! 1 Whe is egative o a factio,! is ot defied. 2. ( 1)[! ( 1)!(2 1) ( 2)!( 1)!] ( 2)! 3. 4. 5.!! ( 1)!!! ( 1)! 1!! 1!! ( 1)( 2)...( 1)!!! ( 1). ( 1)! ( )! 6. 33! is divisible by 15 2 (2!) ( 1)( 2)...(2 1)(2 ) 7. 2 [( 1)!] ( 1)! 8. (! 1) is ot divisible by ay atual umbe betwee 2 ad. 9. 2 (!).! (2 )! Fid, if 10. 11.! 930, 2 ( 2)!!! 20., 5 ( 5)! ( 3)! 12. ( 2)! 60.( 1)! 13. ( 2)! 2550.! 14. 1 1 9! 10! 11!! 15. 990 ( 3)! epaed by: M. S. KumaSwamy, TGT(Maths) age - 1 -
16. ( 1)! 6 ( 1)! 17. 18. 19. ( 2)! (2 1)! 72. (2 1)! ( 3)! 7 (2 )!! : 52 :5 5!(2 3)! 4!( 2)!!! : 2 :1, 4 2( 2)! 4!( 4)! 20. How may 3 lette wods(with o without meaig) ca be fomed out of the lettes of the wod LOGARITHMS, if epetitio of lette is ot allowed? 21. Aditi wats to aage 4 Eglish, 3 Maths ad 2 hysics books o a shelf. If the books o the same subject ae diffeet, detemie the umbe of all possible aagemet. 22. A flag is i the fom of thee blocks each to be coloued diffeetly. If thee ae eight diffeet colous to choose fom, how may such flags ae possible? 23. I how may ways two books of diffeet laguages ca be selected fom 10 Hidi, 5 Eglish ad 7 Saskit books? 24. Fou pesos ete a bus ad they fid seve seats vacat. I how may ways ca they be seated? 25. How may 5-digit umbes ae thee with all distict digits? 26. How may digit odd umbes ca be fomed by usig the digits 1,2,3,4,5,6 whe (i) epetitio of digits is ot allowed (ii) the epetitio of digits is allowed? 27. How may wods ae thee with o without meaig of thee distict alphabets? 28. Thee ae fou outes betwee Delhi ad Mumbai. I how may ways ca a peso go fom Delhi to Mumbai ad etu if fo etuig (i) ay oute is take (ii) the same oute is take (iii) the same oute is ot take. 29. How may 4-digits odd umbes ca be fomed with the help of the digits 1,2,3, 4 ad 5 if (i) o digit is epeated (iii) digits ae epeated? 30. How may odd umbes less tha 10,000 ca be fomed usig the digits 0, 2, 3, 5 allowig epetitio of digits? 31. How may 4-digits umbes ca be fomed usig the digits 0,1,2,3,4,5, o digit beig epeated? 32. How may 3-digit umbes ae thee such that 5 is at uits place? 33. How may umbes ae thee betwee 100 ad 1000 such that at least oe of the digits is 6? 34. How may the thee digit umbes ae thee which have exactly oe of the digits as 6? 35. Fo a set of six tue o false questios, o studet has witte all aswes ad o two studets have give the same sequece of aswes. What is the maximum umbe of studets i the class fo this job to be possible? epaed by: M. S. KumaSwamy, TGT(Maths) age - 2 -
LASS: XI: MATHEMATIS ERMUTATIONS AND OMBINATIONS RATIE QUESTIONS ON ERMUTATIONS FORMULA USED The diffeet aagemet which ca be made out of a give umbe of thigs by takig some o all at a time, ae called pemutatios. Let 1, the the umbe of all pemutatios of dissimila thigs take at a time is give by o (, )! ( 1)( 2)...( 1) ( )! opeties:!, 1!, 0 1 icula emutatio: The umbe of cicula pemutatio of diffeet objects is ( 1)! The umbe of ways i which pesos ca be seated oud a table is ( 1)! The umbe of ways i which diffeet beads ca be aaged to fom a ecklace is 1 ( 1)! 2 ove the followig fo N, 1.! ( )! 2. 1 3. 4. 1. 1. 1 1 1 5. 2. 2 Fid, if, 6. 4 20 2 7. 8. 100. 2 3 2 16. 13. 1 3 3 9. 5 20. 3 10. 11. 12. 13. 30. 2 6 7 1 5 4 : 6 :1 1 4 3 : 9 :1 : 5 :12 1 1 3 3 epaed by: M. S. KumaSwamy, TGT(Maths) age - 3 -
14. 2 1 2 1 1 Fid, if 15. 5 6 : 22 : 7 4 1 16. 15 2730 17. 10 2 9 18. 6 6 4. 1 19. 20 13. 20 1 20. 5 2. 6 1 21. 56 54 6 3 : 30800 :1 22. How may thee-digit eve umbes ca be fomed fom the digits 1,2,3,4,5,6 if the digits ca be epeated? 23. How may 5 digit telephoe umbe ca be made usig the digits 0 to 9, if each umbe stats with 67 ad o digit appeas moe tha oce? 24. I how may ways ca a paty of 4 me ad 4 wome be seated at a cicula table so that o two wome ae adjacet? 25. How may 4-digit umbes ca be fomed with the digits 1,2,3,4,5,6 whe the epetitio of the digits is allowed? 26. How may umbes ca be fomed with the digits 1,2,3,4,3,2,1 so that the odd digits always occupy the odd places? 27. How may diffeet sigals ca be made fom 4 ed, 2 white ad 3 gee flags by aagig all of them vetically o a flag staff? 28. Thee ae how may types of caleda fo the moth of Febuay? 29. I how may ways ca 4 lettes be posted i 3 lette boxes? 30. A boy has 6 pockets. I how may ways ca he put 5 cois i his pockets? 31. I how may ways ca thee pizes be distibuted amog 4 boys whe (i) o oe gets moe tha oe pize (ii) a boy ca get ay umbe of pizes. 32. How may diffeet pemutatios each cotaiig the lette of the wod STATESMAN ca be fomed? 33. Fid the umbe of ways i which the lette of the wod MAHINE ca be aaged such that the vowels may occupy oly odd ppositios. 34. How may wods ca be fomed fom the lettes of the wod SUNDAY? How may of these begi with D? 35. I how may ways ca be lettes of the wod DIRETOR be aaged so that all the vowels ae eve togethe? epaed by: M. S. KumaSwamy, TGT(Maths) age - 4 -
36. The lette of the wod OUGHT ae witte i all possible ode ad these wods ae witte out as i a dictioay. Fid the ak of the wod TOUGH i this dictioay. 37. If the diffeet pemutatios of the wod EXAMINATION ae aaged as i a dictioay, how may wods ae thee befoe the fist wod statig with E? 38. I how may aagemet of the wod GOLDEN will the vowels eve occu togethe? 39. I how may ways ca the wod ENIL be aaged so that N is always ext to E? 40. I how may ways 8 examiatio papes be aaged so that the best ad the wost papes ae eve togethe? 41. I how may ways diffeet books be aaged such that two paticula books ae eve togethe? 42. How may pemutatios ca be made out of the lettes of the wod TRIANGLE. How may of these will begi with T ad ed with E? 43. I how may ways 4 boys ad 6 gils be seated i a lie so that o two boys may sit the eve places? 44. I how may ways 6 me ad 5 wome ca sit i a ow so that the wome occupy the eve places? 45. How may diffeet sigals ca be give with 5 diffeet flags by hoistig ay umbe of them at a time? 46. A oud table cofeece is to be held betwee delegates of 20 couties, i how may ways ca they be seated if two paticula delegates sit togethe? 47. 3 boys ad 3 gils ae to be seated aoud a table. Amog them the boy X does ot wat ay gil eighbou ad the gil Y does ot wat ay boy eighbou. How may such aagemets ae possible? 48. Fid the umbe of ways i which 10 diffeet beads ca be aaged to fom ecklace? 49. I how may diffeet ways ca 5 gil ad 5 boys fom a cicle such that the boys ad gils ae alteate? 50. If 20 pesos wee ivited to a paty, i how may ways ca they ad the host be seated at a cicula table? I how may of these ways will two paticula pesos be seated o eithe side of the host? epaed by: M. S. KumaSwamy, TGT(Maths) age - 5 -
LASS: XI: MATHEMATIS ERMUTATIONS AND OMBINATIONS RATIE QUESTIONS ON OMBINATIONS FORMULA USED Each of the diffeet goups o selectios which ca be fomed by takig some o all of umbe of objects, iespective of thei aagemet, is called a combiatio. The umbe of all combiatios of distict objects, take at a time is give by o (, ). is defied oly whe ad ae iteges such that, 0 ad 0!!( )! opeties: 1 1 0 1, 1 If, the p = q o + q =. p q ove the followig fo N, 1.!!( )! 2. 3. 4. 5. 6. 7. 8. 1. 1 ( 1) 1 1 1 1 1 1 1 1 2 2.[1.3.5...(2 1)]! p q p q o p q 2. 2 1 2 9. 1 1 10. The poduct of k cosecutive positive iteges is divisible by k!. Fid, if 11. 7 5 12. 10 15 epaed by: M. S. KumaSwamy, TGT(Maths) age - 6 -
13. 30 4 14. 2 3 3 : 11:1 15. 3 6 3 : 33: 4 16. 2 3 2 Fid, if : 12 :1 17. 15 15 3 3 18. 8 7 7 3 2 19. 18 18 2 20. 15 15 1 : 11:5 Fid ad, if 21. 1 36, 84, 1 126 22. 1 : : 1 3: 4 :5 23. 1, 1 24. : 11: 6, : 2 :1 1 1 1 1 25. : 1 : 2 1: 2 : 3 26. I how may ways ca 11 playes be chose ou of 15 if (i) thee is o estictio (ii) a paticula playe is always be chose (iii) a paticula playe is eve chose? 27. Out of 5 me ad 2 wome, a committee of 3 is to be fomed. I how may ways ca it be fomed if at least oe woma is to be icluded? 28. A committee of 5 is to be fomed out of 6 me ad 4 wome. I how may ways ca this be doe, if (i) at least 2 wome ae icluded (ii) at most 2 wome ae icluded? 29. Thee ae poits o a cicle, fid the umbe of (i) lies which ca be daw (ii) tiagles which ca be fomed. 30. How may diagoals ae thee i a polygo of sides? 31. A polygo has 35 diagoals. Fid the umbe of its sides. 32. I how may ways a goup of 11 boys ca be divided ito two goups of 6 ad 5 boys each? 33. Fo the post of 5 cleks, thee ae 25 applicats, 2 posts ae eseved fo S cadidates ad emaiig fo othes, thee ae 7 S cadidates amog the applicats. I how may ways ca the selectio be made? 34. I how may ways ca 10 diffeet books o Eglish ad 5 simila books o Hidi be placed i a ow o a shelf so that two books o Hidi ae ot togethe? 35. I a examiatio, a cadidate has to pass i each of the 5 subjects. I how may ways ca he fail? epaed by: M. S. KumaSwamy, TGT(Maths) age - 7 -
36. A questio pape has two pats A ad B each cotaiig 10 questios. If a studet has to choose 8 fom pat A ad 4 fom B, i how may ways ca he choose the questios? 37. A cicket team of 11 playes is to be selected fom 16 playes icludig 5 bowles ad 2 wicket keepes. I how may ways ca a team be selected so as to cosist of exactly 3 bowles ad oe wicket keepe? 38. Thee ae 15 poits i a plae, of which 6 ae colliea. How may (i) staight lies (ii) tiagles ca be fomed by joiig them? 39. A bag cotais 4 ed, 3 white ad 2 blue balls, thee balls ae daw at adom, detemie the umbe of ways of selectig balls of diffeet colous. 40. A ma has seve fieds. I how may ca he ivite oe o moe of them to a paty? 41. A fathe with 8 childe wats to go to zoo as ofte as he ca without takig the same thee childe moe tha oce. How ofte will he go ad how ofte will each child go? 42. Thee ae two oud tables oe with m seats ad the othe with seats aoud it. I how may ways ca (m + ) guest be seated at them? 43. Fom 7 cosoats ad 4 vowels, how may diffeet wods ca be fomed cosistig of 3 cosoats ad 2 vowels? 44. To fill 12 vacacies, thee ae 25 cadidates of which 5 ae fom S caste. If 3 of the vacacies ae seved fo S cadidates while the est ae ope to all. Fid the umbe of ways i which the selectio ca be made. 45. Fom a class of 10 boys ad 6 gils, 10 studets ae to be selected fo a competitio, at least icludig 4 boys ad 4 gils. The 2 gils who wo the pizes last yea should be icluded. I how may ways the selectio ca be made? 46. A committee of 5 is to be selected fom amog 6 boys ad 5 gils. Detemie the umbe of ways of selectios if the committee is to cosist of at least oe boy ad oe gil. 47. A questio pape cotais 12 questios divided ito 3 pats, pat A cotais 6 questios while pat B ad cotais 3 questios each. A cadidate is equied to attempt 6 questios selectig at least oe fom each of pats B ad. I how may ways ca the cadidate select 6 questios? 48. A bag cotais 4 ed, 2 white ad 3 blue balls, thee balls ae daw adom, detemie the umbe of ways of selectig balls at least oe black ball to be icluded i the daw. 49. A cadidate is equied to attempt 6 out of 10 questios, which ae divided ito goups each cotaiig 5 questios, ad he is ot pemitted to attempt moe tha 4 questios fom each goup. I how may ways ca he make up his choice? 50. A cadidate is equied to attempt 7 out of 12 questios, which ae divided ito goups each cotaiig 6 questios, ad he is ot pemitted to attempt moe tha 5 questios fom each goup. I how may ways ca he choose 7 questios? epaed by: M. S. KumaSwamy, TGT(Maths) age - 8 -