Pp, ="P then r= et then a= 3 279 —3)3/4 yay (20m, = 3: Sthen n= 2)5 3)6 4)3 | DS, 32760, then r= number of different signals can be 1h venby using any number of flags from 4 ai of different colours is Mrenumber of words that can be formed Ih ing any number of letters of the word KANPUR" is m0 21956 3360 4)370 ‘Three Men have 4 coats 5 waist Coats, Ie ud caps. The number of ways they can swear them is 8p, 2) 4°5°6_3)P,$P,°P,4) 180 ‘Arailway carriage can seat 5 each side. The number of ways a party of 4 girls and 6 boys can seat themselves so that the girls may always have the corner seats is 1)17,430 2) 17,431 3) 17,280 4) 17,281 48, The number of words that can be formed using all the letters of the word "KANPUR" when the vowels are in even places is 1144-2) 36 3)24 4) 48 41, The letters of the word "LOGARITHM" are arranged in all possible ways. The number of arrangements in which the telative positions of the vowels and consonants are not changed is 2. The number of ways one can arrange words with the letters of the word "MADHURI" ‘0 that always vowels occupy the 1g, middle and end places is a 2)9C,4P, 3)34C, 4) 3141 letters of the wd re OER are 4 at a time and arranged in all Ways. The number of C which begin with 'F' and end 218 314412 Are-to be seated in a row of three are ladies. The ladies insist together while two of the 26. 27, 28. 29. 30. PERMUA 31. 32. Seats. In how many ways be seated? er. 1) 2399976 2)218446 0 3) 630624 4) 181440 Ifa? denotes the number of : of +2 things taken all at a time, b the number of permutation of x things taken — 11 at a time and ¢ the number of Permutations of ~ 11 things taken alll at a time such that g = 182hc, then the value of xis ; The number of different ways that three distinct rings can be worn to4 fingers with atmost one ring in each of the fingers is )'R 2)*R” 3) Raa Six examination papers are to be set im a certain order not to be disclosed. It is” discovered that one order has been leaked out. The number of ways that their order can be changed is 9 articles are to be placed in 9 boxes one in each box 4 of them are too big for three of the boxes. The number of possible arrangements is 19! 2)514! 3) *Bx514)5t6t. The number of permutations of ‘n?_ dissimilar things taken 'r' at a time, im which a particular thing always occuris DP, 2) 0)", 3)n“5P. oye, If words are formed by taking only 4: time out of the letters of the PHYSICAL, then the number of word which Y occuris 2 1 a5 ae ATIONS OF DISS} THINGS | ‘The number of ways in which Girls can sit in a row so maybe together is 1615! 2)6!7P, 3)K ‘The number of ways women be seated in a 1) *p, _ the boys and the wé is 1) 41312! 341 GY) 2 q34, 35. 36. 37. 38. 39. 40. 4l. MUTATIONS & COMBINATIONS ich 10 books can nber of ways in whi be arranged in a row such (hat (we specified books are side by side Is 10! a oa 29! 3) 9121 4) > A family of 4 brothers and 3 sisters Is (0 be ed ina row fora photograph. The number of ways in which they ean be seated if all the sisters are to sit together is 1) 120 2)240 —3)360 «© 4) 720 ‘The number of ways in which the time table for Monday be completed if there must be 5 lessons that day (Algebra, Geometry, Calculus, Trigonometry, Vectors) and not Algebra and Geometry must immediately follow each other are 1) 72 2)5! 3) 3x 51/2 4) 6! ch 5 Boys and 5 a row so that no ways in wh nged th two girls 1) 10! 3)(51P 4) There are 10 white and 10 black balls marked 1,2,3..... 10. The number of ways in which we can arrange these balls in a row in such a way that neighbouring balls are of different colours is 1) 10!9! 2)20! 3) (10!)? 4) 2(101) The number of ways in which 10 candidates 51)? Ai Ayy Ay A, sone sA,, can be ranked if A, is just above A, is 1) 9! 2! 2) 10! 3) 10!2! 4)9! The number of ways in which the candidates A,,A,,....,A,, can be ranked if A, is always above A, is , 10! 1912! 2)9! 3) 4) 10! A, B, C are thre 7 persons who speak umber of ways in which it can be done if'A' speaks before 'B' and 'B! speaks before 'C' ig 1)820 _2)830 3) 840-4) 850 The number of permutations altogether of n things when r specified things are to be in an assigned order, though not necessarily consecutive is nt nt a 45. 46. 47. 48. 49, 1) 12) (r—ryint Dp Von) at 42. 43. 44. ‘The number of four digit can be formed with 1, 2, 1)4.'P, 2)5.5P) aie ‘The number of four digit can be formed so that repeated in any number is 1)2240 2)2420 3) The number of numbers of§ zero digits such that all d first four places are less ths the middle and all the digits places are greater than the dj middle is 1241) 2 (41 3) BF The number of four digit nu be formed with 0, 1, 2,3, 4,5i5 WP, 25. 3) P, - 2P, 4), ‘4 ‘The number of four digit. can be formed with 0, 1, 2,3, 1) 180 2)175 3) 160 Number of 6-digit telephone which can be constructed: 2,3, 4, 5, 6, 7, 8, 9, ifeach with 35 and no digit appears is 1) 1680 2) 8! 3)6! The number of numbers lyit and 1000 that can be formed) 4, 5, 6 so that no digit being any number is 1100-2) 125 The letters of the word nged in all possible of arrangements in wh letters between R and Di 3) SUNDAY" withoutb Without ending with 'Y' 696 2) 624BILITY WITHOUTREPETITION siaumber of seven digit numbers a sible by 9 formed with digits 1,2, 3,4, 4 M7, 8.9 without repetition fy yt! 2°R, 3)3(7!) 4) 4(71) ‘number of digited numbers which are divisible by Sand which contains of s ad digits i my 2) 120 ‘hesum of inte divisible by 2 or Sis 13000 2)3050 3) 3600 4)3250 The number of ways in which we can arrange the digits 1 2,3 such that the product of five digits at any of the five onsecutive positions is divisible by 7 is 7! 2)°P, —-3) 8! 4) 5(71) _ If repetitions are not allowed, the number of numbers consisting of 4 digits and divisible by 5 and formed out of 0, 1, 2, 3, 3) 24 4) 32 m 1 to 100 that are ss. igit number divisible by 6 is to be formed by using 0, 1, 2, 3, 4, 5 without repetition. The number of ways in which this can be done is 3)108 4) 216 visible by 3 is to be formed using the digits 0, 1, 2, 3, 4, 5 without repetition. The total number of ways this can be done is 1120-2) 96 3)216 4) 220 . The number of five digit numbers that be formed with 0, 1, 2, 3, 5 which n are TON igits at the ten’s place of allthe numbers formed with the digits of ie 5,6 taken all at a time is thee, 2108 3364) 18 fe ‘um of the value of the digits at the the laa of all the numbers formed with The nh 0f3, 4, 5, 6 taken all at a time is formed b of all the numbers that can be Sig Y ‘King all the digits from 2, 3, 4, a 2) 79,992 4) 78,456 63, 65. 66. | 67 If the letters of word “VICTORY? are { 69 The letters of the word "DANGER" are. 70. 7. ‘The sum of all 4 digited even numbers that can be formed from the digits 1, 2,35 4:5 is 1) 1,58,994 2) 1,59,984 3) 1,59,894 4) 1,59,884 ‘The number of five digit numbers that canbe formed with 0, 1, 2,3, 5, which are divisible by 5 is 18. IL: The sum of all four digit numbers that can be formed with 1, 2, 3, 4 so that no digit being repeated in any number is 66660. which of the above statements is true? 1) only! 2) only IL 3) Both I and Il 4) Neither I nor Il ‘The sum of all the four digit numbers that can be formed with 0, 2, 3, Sis 1) 66660 2) 66480 3) 64440 4) 65520 ‘The sum of all 4 digited numbers that can be formed by taking the digits from 0, 1, 3, 5,7, 9 is 1) 16,23,300 2) 16,32,300 3) 16,32,030 4) 16,33,200 RANK OF THE WORD arranged in the order of a dictionary. Then rank of the word ‘VICTORY’ is )3731 2)3732 3)3733-4)3720 The rank of the word "MADHUR" when arranged in dictionary order is 1)362 2360-3) 358 4356 permuted in all possible ways and the words thus formed are arranged as in a. dictionary. The rank of the wo "DANGER" is 1) 132 -2)133. 3) 134 4) 135 All the numbers that can be formed usin, the digits, 1, 2, 3, 4 are arrang increasing order in magnitude. of the number 3241 is 1) 56 2) 16 3) 55 All the numbers that can the digits 1, 2, 3, 4, S are decreasing order of ma of 34215 is 1) 58 2) 6272. fofe VT TN edb The letters of the following words are arranged and words thus formed are kept as in dictionary, Then arrange the following in the descending order of their ranks A:SITA B;RAMUC: TEA D: BUT 1)DCBA 2) ABDC 3) ABCD 4) BACD PERMUTATIONS WHEN REPETITIONS TB. 74. 7S: 76. D-There are 5 multiple choi AREALLOWED Ifthe number of ways in which n different things can be distributed among n persons so that at least one person does not get any thing is 232. Then nis equal to 1)3 24 3)5 4)6 A man has3 servants. The number of ways in which he can send invitation cards to 6 of his friends through the servants is 1) 36 26 In a steamer there are 12 stalls for animals and there are horses, cows and calves (not less than 12 each) ready to be shipped. In how many ways can the ship load be made such that any stall can’t have morethan one animal 6 , 2S es 1) 3-1 2) 3! 3) (12)'-14) (12) The number of ways of wearing 6 different rings to 5 fingers is a) 5% 2) 6° 3)(5? tae)" questions in test. If the first three questions have 4 choices each and the next two have 5 choices each, the number of answers ossible is 8Sisteen men compete with one another in running, swimming and riding. How many prize lists could be made if there were altogether 6 prizes of different values, one for running, 2 for swimming and 3 for riding 1) 16x 15x 14 2) 16° x15? x14 3) 16° «15x14? 4) 16? 15x14 79. The sum of all 3 digited numbers that can be formed from the digits 1 to 9 and when the middle digit is a perfect square is (repetitions are allowed) 1)1,34,055 . 2)2,70,540 3)1,70,055 4)2,34,520 adhe 80. 81. 82. 83. 84. 86. 87. 88. 89, 90. . The number of one one functions! ‘The number of numbers forn more than 6 digits using the 5 when repetition is allowed j 1) 1092 2)1090 3) 1085 The number of permutatig dissimilar things taken more than time when repetitions are all exceeding 25) Berea Diemer) 2 Fes 25 (5.525 15 25 1 3) 54(25 +25") Number of 5 digit palindromes is 1) 8100 2) 900 3) 90000 Number of 5 digit Even palind 1)400 2)500 3) 4000 Number of functions from Set-A eg 5 elements to a set-B contai elements is 1) 5* 2) 4° 3) 4! can be defined from 4 = {a,b,c} int B= (1,2,3,4,5} is 1) 5P, 2) 5c3.-» Sil The number of many one funeti A={1,2,3} to B={a,b,c,d} is 1) 64 2)24 3) 40 The number of onto functions that defined from A = {a,6,c,d,e} to B={1 1) 30 2)0 3) 60 ) The number of one one onto functi ed from A=(a) can be defi B={1,2,3, 4} The number of constant map A=(42,3,4.........,n} to B= {ab} In! 2)n 3)2 CIRCULAR PERMUTAI ‘The number of ways in wh girls can sit around a table: girls do not come together + The number of ways that: with 18 flowers such th ea d9. a 95. 96. 7. 98. 99, 100. fe number of Ways in which 5 men, 5 ; Masits and 12 children can sit around a Mreular table so that the children are always together is ata 12! 2) 111 121 3) 10! 12! 4y(i2ty 30 persons are invited for a party, The different number of ways in which they can be seated at a circular table with two particular persons seated on either side of the host is 1) 19!2! 2) 1812! 3)20!2! 4) 1813! ‘The number of ways in which 4 men and 4 women are to sit for a dinner at a round table so that no two men are to sit together is 1) 576 2) 144 3) 36 4) 120 The number of ways in which 7 men can sit at a round table so that all shall not have the same neighbours in any two, arrangements is 7 women and 7 men are to sit round a circular table such that there is a man on either side of every women. The number of seating arrangements is Dy —-2)(6)?_ +3617! 4) 7! The number of ways that a garland can be made out of 6 red and 4 white roses of different sizes, so that all the white roses come together is The number of ways in which 7 men be seated at a round table so that two particular men are not side by side is The no, of ways in which 6 gentlemen and 3 ladies be seated round a table so that every gentleman may have a lady by his side is ... 11440 2)720 = 3) 240-4) 480. CONSTRAINT PERMUTATIONS ‘The number of words that can be formed from the letters of the. word "INTERMEDIATE" in which no two vowels are together is op, 1)61.7P, 2) aaaial 6p, cy iat 4) ade out of the tet he wo "ENTRANCE" so that Noe always together is “ n Day. B17 ye oe ‘The number of permutations that e made by using all the letters of the: TATATEACUP that start with A and with U is 1) 5040. 2) 8151313! 3)3360 4): }. Number of ways of | \uting the letters of the word "ENGINEERING" so that the order of the vowels isnot changed is it 2) ee 102. Dep, uu! up 3) 5151 |. The number of different numbers tt cs be formed by using all the digits 1, 2, 3, at odd digits always occupy the odd places is 5. A three digit number n is such that the last . The number of ‘n” no two consecul 1) 91 2) at }. The number of five 1) 84 2)90 3)72 4) 60 . The number of numbers greater equal to 1000 but less than 4000. be formed with 0, 1, 2,3, 4 so that: may be repeated is . The number of permutations th made out of the letters of the ¥ "MATHEMATICS" When come together is BL.4t » 2 a oT isThe number of permutation that can be je out of the letters of the word LATHEMATICS", When no two vowels come together is 1 ey v2 1) 7!"P, ay 2! . The number of permutation that can be made out of the letters of the word "MATHEMATICS" When the relative positions of vowels and consonants remain unaltered is SEF 22.71 3)7! 44-7 3. In the word 'ENGINEERING’ if all ‘E”s are not together and N's come together then number of permutations is o 7 lam or 1 The number of ways in which the letters of the word MULTIPLE be arranged without changing the order of the vowels is 1)3360 2)20160 3)6720 4) 3359 f ne N and 300 r=4 Be Be | 0 78)2 79)1 80) | 12, "P+r"p, =p ml 82283) 1 84)2 "p, "pp *p a 86)3 87)1 88) 24 13. ible . Bs 90) 4320 91)4 92)3 5 ie = yz 94)2, —-95)36096)3 = 6 =a(b+c)>9=axT yp 86t0 98) 480 99) 1 100)2 oy3102)3. © *103)4——104) 18 Ma {053 106)2-«:107)3.—— 108) 2 jooy375 1104 NN) 112)1 34 ~=—«*1N4)1 115) 539 116) 4 : pint —-*118)531119)3 120) 1 15. 32760= 15x 14x 13x12 >r=4 pl ipeaienosye 124) 4 16, “R442 442 ee 1, By fundamental theorem of addition | 17 ‘P+ %P WP, * Pat She ie : 0+8=18 18. Wearing of coats =*P, } Wearing of waist coats=§P, 2 AG i Wearing of caps ="P, 7 3, Byfimdamental theorem of Multiplication=5x8 | 9 . AIx6! lamer ‘orem of Multiplication= 4, Borealis ultiplication= | 55, > vowel oociny 3 Eve RE remaining 4 places ocupy by.co1 5, +214 31+ 414 51-153 -. Ans*P, x41 6. n!=1.2.3.....nie. n! is divisible by any number | 21. LGRTHM; OAI- 6!.3!=4320: between 2ton. .. n! +1 when divided by any } 22. 3 vowels, 3 places mumber between 2 to n leaves ‘1’ as remainder Total=314! ts 23 1 Qel=11 21 31 4! 5! 6! 718! 9110! | 24, ct Mt... 25. Pe 2 digits 01 02 06 24 20 20 40 20 80.00 | 96, 4x34 ea ; 27. 6!-1=719 , +, lasttwo digits in above expression are same | 55 4 big. artieles niseed aetna atheist two digits of Dy A!=13 Beep 8. (n+1) - 1) n! = (n+)! )” kind, There are in all 1p objects : ()! themumber oftheir arrangements is nly isaninteger. is divisible by (7!) Add 1!+2!+3!+4!+5! find the digit in 10's place, it is 1 (From 10! +9! and onwards the units and 10’s places is zero only). 4 Fromsynopsis Il statement is true §. The given 6 digits can occupy the numerator and denominator places in ° P, =30 ways. But 1 outof them — represent same number 246 24 1’3°3 TePresent same number. 5+% 36 number. Similarly 5>% r 12,36 epresent same number , 3°6 present same number, In all above cases We Neto consider only one case each. * No. of rational numbers O30-(2424141+1+1H 2+ 1 1" isa rational no) =23 6. 10. 4 Ts I letter is posted in 5 ways => 5* II: The given word is PHYSICS 1) Alldifferent along with ¥ = ‘C, x41= 240 2) Two same along with ¥ = Total = 288 Required number 9! " si | Tt T= [+35 Bt= 16s A= 4px (14244 +5+6)<1191 = 279952 I: 3x5143%4140%3!+ 2x 214121 1 = 438 =618 Ik: 5x 51+0x 4142 «31+ 22hetx te Total variables if only alphabet is used = 26 ¥ : Total variables if alphabets and digits both are used = 26.10 286. => Total variables = 26(1+10) The first entry is a letter. The other entries are, letters or digits. The desired number is 26 s =< (36° -1) > 26.36" = The possible sizes of matrices haveing 4 elements are 1x4, 4x1, 22; (ie 3 ways) 4 placess in each matrix can be filled in 4# ways No. of different matrices = 3x 4# Formed number can be atmost ofnine digits. Total number of such numbers =3437 +3? tenet 3? $23" No.ofnumbers = Total Possible numbers: using all digits - Total possible numbers v out ‘1’-1 =10"-9"—1AL ocbualond EXERCISE -1 JNITIONS PROPERTIES of or "C, | sarsanen °C 2) may be fraction aninte ps Et inber 4) an even number isn : GC. C, then x= hs r-1 3)n 4)r+1 f Thevalue of 1x35... 2n-1) 2"= n)! n! 4)2n Reset contains (2n + 1) elements. If the pumber ofsubsets of this set which contain ttmost ‘n’ elements is 4096, then the value ofnis jnalibrary there are (2n+1)books. If a student selects atleast (n+1) books in 256 yays then the number of books in the library is 2)8 3)9 4)6 & Itc, ="°C,,, then r= 2)3 3)4 4)5 )5 ) ) 1, IPC, :*C,=44 : 3 then n = { 1fCQ2n,3): C(n, 2)=12: 1, thenn= 4-25 3)6 4)8 4% “C,+>)(18- /)C WC. Dy tC. 3) 4c, 4) "C, 1 I 1:The no. of 3 digit numbers of the form Ayewhere x>y>z is 1. Il; The no.of 3 digit numbers of the form WZ where x>y>z is "C,3! Which ofthe above statement is correct? Monty 2) only Il )Both and 11 4) Neither I nor II father with 6 children takes 3 at a time without taking the same children. Often father goes to the park? 2) 16 3) 18 4) 20 20. 21. TIPE ew ate hoo » No. of ways of selecting 2 girls and 3 boys from 3 girls and 5 boys is 120 2)24 330 48 . The number of subsets of the set A= [1,2,3,....9) containing at least ome oddnumberis 2°) - 2 . No. of ways of selecting 3 consecutive objects out of 15 objects (distinct) See eg around acircleis ( W 1 *“ Niteq. aya, re * 4) 15 . The number of quadratic expressions with the coefficients drawn from the set {1,2,3,4}is ave Sonal auaees . A student is to answer, 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number es clots available to him is 1)196 2)280 3) 346" nae 140 Ten students are participating in a race. ‘The number of ways in which the first three places can be taken as 13 2) UCR See 4) 9s n a chess tournament, where the participants were to play one game with another. Two chess players fell ill, having played 3 games each. If the total number: of games played is 84, the number of participants at the beginning was . From a company of 20 soldiers any 5 are placed on guard, each batch to watch 5 For what length of time in hours can ferent batches be selected is 2) *P, 3) 4C.x5 4) *P,x5 10 persons are seated at round table. The number of ways of selecting 3 persons out of them ifno two persons are adjacent to each other is 150 2)62 3) 56 4)57 Out of 9 boys the number to be taken to form a group, so that the number of different groups may be greatest is nae 2)5 3)40r5 4)622. A train going from Vijayawada to itaeeabea i ‘at nine intermediate Stations. Six persons enter the train during the journey with six different tickets of the same class. The no. of different tickets they may have DG ee) C st ayy 4) "C, A party of 9 persons are to travel in two Vehicles, one of which will not hold more than 7 and the other not more than 4. The number of ways the party ean travel COMBINATION OF nD AR THINGS IN WHICH or THINGS ARE UDED or EXCLUD! From 15 players the number of ways of selecting 6 so as to exclude a particular player is ee Cae) Cee, . The number of all th Jement subsets of the set {a,, a,, a, ,.....,a,} which contain BeaGs eG 26. Ina shelf there are 10 English and 8 Telugu books. The number of ways in which 6 books can be chosen if a particular English book is excluded and a particular Telugu book is excluded is 1)°C,.7C, 2)"*C, 3)°C,.*C, 4) ¥C, . The number of ways in which a team of 11 players can be selected from 22 players including 2 of them and excluding 4 of them is Ae 3)"C, 4) ; The number of permutations of n thin taken rat time if 3 particular thi always occur is 2)" (1-3)! ( 3 3)! r(r-1)(r-2) 2) (n 4) - . Out of 7men and 4 women a committee of 5 is to be formed. The number of ways in which this can be done 1: 50 as to include exactly 2 women is 219, TI: so as to in Which of the 1) only! 3) Both Land It 30. A guard of 1S mens of n soldiers. The n particular soldiers: -(n- 3) [31(n-15)! By A father with 8 children time to the zoological gard he can without taking the sam together more than once, of times a particular child Zoological garden is COMMITTEE PROBI | 34 group contains 6 men committee is to be formed containing 3 men and 2 number of be formed is 7 °C, 27°C, x3¢) ayaa 33. A team of 11 players has to from the groups consistin players respectively. Thent of selecting them so that contains atleast 4 players fi group is 1) 120 34. A commi 31. 2) 280 3) 344 two particular women are: in the committee, th committees fo 1) 420 3) 336 4) 21 35. A reserve of 12 railways to be divided into two gre forday duty and theo The number of was done if two specified pe Not be included in thes 1)300 2)504doc Fenmittee of 6 18 chosen from 10 men A 7 women so as to contain atleast 3 men E K 2women If2 particular women refuse - serve on the same committee, the ‘umber of ways of forming the committee is ‘ v crew of an 8 oar boat is to be choosen 12 Men, of whom 3 can row on strok Ea only. If selected the n 9 MYA ewe i formed using these points as yertices s ; 1)205 2220 += 3) 225-4) 230 45. There are 'p' points in space of which '4” Points are coplanar, Then the number of » planes formed is ly mber of ways) 1) °C ~8C, Cth ! fheerew can be arranged is Ri 3)"C, -4 4) °C, -9C, | % ~ F Rec, to 2) 14x91 1 46, ABCD isa convex quadrilateral 3,4, 5 and 4 q 6 points are marked on the sides AB, BC, ! ¢,%¢)-4!.4! 4) 2080, 41.41 CD and DA respectively, the number of 4 scarwill hold 2 persons in the front seat 48 {hd Lin the rear seat. If among 6 persons aly 2 can drive, the number of ways, in ‘which the car can be filled is pio 2) 18 3) 20 4) 40 yf a plane there are 37 straight lines of which 13 pass through the point A and 11 pass through the point B. Besides, no three lines pass through one point, no line passes through both point Aand B, and no twoare parallel. Then the number of points ofintersection of the lines is Mi Ahe greatest number of points of jee of 8 lines and 4 circles i ( Maximum number of point of intersection made by 5 circles and 3 triangles 70 2) 80 3)128 4) 100 4, Ifaline segment be cut at ‘n’ points, then the number of line segments formed is 3) n(n 1)n(n+3) 2) pet 2hn+ 6) 2 ©. There are 10 straight lines in a plane no {Wo of which are parallel and no three are Concurrent. The points of intersection are , then the no. of fresh lines formed 4yn 2) 615 0 4) 600 sides AB, BC, CA of a triangle ABC 3,4 and 5 interior points respectively The number of triangles that can triangles with vertices on different sides is 1)270 2)220 3) 282 4) 342 47. There are ‘m’ points ona straight line AB and ‘n’ points on another straight line AC in which Ais not included. By joining these points triangles are constructed. i) When ‘Ais not included. ii) When A is included, the ratio of number of triangles in both cases is m+n-2 m+n-2 m+n 2 m+n-2 m+n+2 m+n+2 m+n-2 48. There are three coplanar lines, if any m points taken on each of the lines, the maximum number of triangle with vertices at these points is 1) m? (4m-3) 2) 3m? (m—3)+1 3) 3m’ (m-3) 4) m(4m—3) 49. In a polygon no three diagonals are concurrent. If the total number of points ofintersection of diagonals interior to the polygon is 35 and the number of diagonals is 'x', number of sides is "y' then (y, x)= 15,5) 2)(6,9) 3) (6,20) 4) (7, 14) if m parallel lines in a plane are interseete¢ by n parallel lines then number 0 Hlelograms formed is parallelograms formed ow min! Yay ») (n=) a a 3) @yon-2lo-2)!52. 53. 54, - No. of ways of distri Sl. The number of ways of selecting two | 58 1 “Squares (1x1) ina chess board such that they have a side in common is 1224 ay 112 3)56 4) 68 The number of rectangles which are not Squares in a chess board is ->8 DC, xtc ig 2) (8,, x8, 4) (9, x9,)-E8* ‘arked on the circumference 2 circle at equal distances. Then the 3)°C, x°C, - g2 8 points are m: of them is Pee he number of number of squares can be drawn by joining an * 1)*p, Zee 3) — 4)2 2 DISTRIBUTION INTO GROUPS The number of ways in which 52 cards can be divided into 4 sets of 13 each is 52! Sat. _ 52! aay 2 Fag 9 a 9 ia ibuting (p + q +r) ‘nt things to 3 persons so that one Person gets p things, 2nd person q things 3rd person r things is differe (p+q+r)! = (p+qtr)! ) x3 i plair! plqls (p+q+r)! bee PY a pigir! 4) platy 56. 15 Passenge: 57. Ts are to travel by a double decker bus which can accomodate § in upper deck and 10 in lower deck. The number of ways that the passengers are distributed is A class contains 4 boys and ‘g’ girls, Every sunday five students, including at least three boys go for a picnic to Zoo Park. a different group being sent every week. During, the picnic, the class teact each girl in the group a doll. If number of dolls distributed was value of *g’ is is 2)12 her gives the total 85, then 3)8 4)s objects into 3 Il. The no.of ways in distributed among 4 52! @y Which of the above stat be formed with 8 prime num 1247 -2)252.-3)5 No. of ways of selecting none 10 identical things is yr 210 ay | basket contains 4 Oranges, §, 6 Mangoes. The number of wayga make selection of fruits from th is A) 209 2)210 320 he no of ways of selecting al letter from the letters of th “PROPORTIO! 63. Atan el number of number to be candidates and 4 are to be el i Voter votes for atleast one candidal the number of ways in which hee is 385 2)1110 3) 5040 0(k =1,2,3,4.5) is 1) 300 2)350 3) 336 4) 316 81, Inhow many ways 3 boys and 15 girls can sit together in a row such that between 2 boys there is atleast 2 girls 1) 14,3115! 3) 18,3115!PERMUTATIONS & COMBINATIONS 82. Five distinct letters are to be transmitted through a communication channel. A total number of 15 blanks is to be inserted between the the letters with at least three between every two. The number of ways 1) 1200 2) 1800 3) 2400 4) 3000 83. The number of positive integral soluti of x° — y? = 352706 is 2 21 3)0 4)3 84. If N is the number of positive integral solutions of x,x,x,x, = 770, then N= APPLICATION OF ONTO FUNCTIONS 85. Theno.of Sdigit numbers that can be made using the digits 1 and 2 and in which atleast one digit is different is 86,\4 balls of different colours are to be kept in 3 boxes of different sizes. Each box can >, hold alll five balls. Number of ways in which ‘the balls can be kept in the boxes so that ‘no box remain empty is '87. The number of ways of distributing 9 identical balls in 3 distinct boxes so that none of the boxes is empty is Dec 2) 28 3) 8 4)5 88. In the shop there are five types of i creams available. A child buys six ice- inwhieh thiscanbedoneis ‘il jons creams. Statement - 1 : The number of different ways the child can buy the sixice- creams is}! Statement - 2 : The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging 6A’s and 4 B’sin a row 1) Statement-I is true, Statement-2 is true and Statement - 2 is not a correct explanation for Statement - 1 2) Statement-1 is true, Statement-2 is false 3) Statement-1 is false, Statement-2 is true 4) Statement-1 is true, ‘Statement-2 is true and Statement-2 is a correct explanation for ‘Statement-1 89. 91. 92. The no. of permut the word ‘PROPOR’ time so that 3 are alike is 115 -2)20 How many different out of the letters "MORADABAD" taken 1)620 2)622 3) 626 MISELLANEOUS P ‘The number (36)! is di 1) 2% 2) 7 (12) | The number of divisors of. a remainder of 1 when di KEY 1)4 2)4 3)Z9 5)3 6)2 2 _ 93 10)4 13)496 144 17)3 18) 15 21)3:. i 22 25)3 26) 293 —-30)3 333. 34)3 36 ) 7800 39) 535 40) 104 43)1 44) 47) 1 48) 1 512 $2)4 55)1 $6) 3003 59)1 60)3 63)1 64)3 67)4 68) 71)4 72) 120 75) 4 76) 1 79) 224 80)3 83)3 84) 256 872 -88)3 913 92688. 89, 90. 91 . Divisors are of the form 4m +1 87. 3x4C, «7G, x'C, +3x°C, x°C,*'G, n™—"C,(n-1)" +" C,(n-2)"— Where n=3 m=5 ” ; The problem is equivalent to distributing identical thingsto 3 persons so that each. eer can get at least one thing ie. Number 0} Positive integral solutions of x, +, +) =9 =>'C, =28 x20 XM FX) +x, +x, +x, = 6 64541, ig 2 Ao! No. of solutions = “**'C,_, ='"C, 61a! 3 alike =3 O's=1 | different from {P, R, T, I, N} =SC, 4! eit MORBAD AD A 4diff: °p, = 360 Pat al 2 diff+ 2 same : *C,.*C,.5° = 240 4! 5), 20 31 21.2 3 same + | diff 2 same + 2 same ; =6 626 £2 in 361 is FH EH es) 2314] 8 |*Li6] 32 =18+9+4+24+1 Exponent of 3 in 361 is 36], [36], [36 3 tl o |tla7| =12+44+1=<17 36! = 35.3" x50, Maal? 3 5A | 36! is divisible by "” they are find the when divided odd divisors. Among odd divisors divisors which leave remainder | by4 1, A eh. PofA is chosen. The set A is era 21, 22, 4 [5 The results of 21 football matches ( a he number of the interval (2002, 2003) ‘er the decimal point ar ire in decreasing order is -) ye 22 3) 94 a set containing n elen by replacing the elements of P. of Ais again chosen. The numb of choosing P and Q so that PQ exactly two elements is 1)9.*C, 0 ; 4) °C, 368 ere are 10 bags B,, By, BysenuBy > sy 30 different ay respectively. The total number ofy bring out 10 articles from any one ba 1) "Cy-"Cy 2) *C,, 3): A Tennis tournament is to be played by} pairs of students and each pair iso) with every other pair one set. Iffours are played each day then the numberdl days should be allowed for the tot is 1) 12 2) 16 3) 80 4)90 lose, draw) are to be predicted. The! different forecasts that can contain 19 is bit strings are made by filling thed 9or1. The number of stringsin Wh tly k zeros is ec 2) orn The number of ways in which we ci 4 numbers from 1 to 30 so as tot ery Selection of four conse Numbers is 1) 27378 3) 27504 m 1, 2, 3, -17/2 3)(n+ 12/2‘A child attempts to open a (each disc consists ordight (Oy aan He takes 5 sec time to dial a particular number on the disc. If he does so for's hrs every day, then the number of days he would be sure to open the lock is The number of ways of selecting 3-member subset of {1,2,3,.....25} so that the numbers form a GP. with integercommon ratio is N10 -2)11 3) 12 4) 15 11, The number of n digit numbers, which contain the digits 2 and 7, but not the digits 0,1,8,9. 1) 6" —5"+4" 2) 6" 45" 5" 4.4" 3) 6" +5" -5"-4' 4) 6 5" — a4 12. Let N be the number of 4 digit numbers formed with at most two distinct digits. Then the last digit of N is 16 2)7 3)8 4)9 13. Two teams are to playa series of 5 matches |" between them. A match ends in a win or loss or draw for a team. Anumber of people forecast the result of each match and no two people make the same forecast for the series of matches. The smallest group of people in which one person forecast correctly for all the matches will contain ‘n’ people, where n= 1)81 2) 243 3) 486 4) 144 A delegation of four friends are to be selected from a group of 12 friends. The number of ways the delegation be selected if two particular friends refused to be together and two other particular friends wish to be together only in the delegation. A boy has3 library tickets and 8 books of his interest in the library. Out of these 8, he does not want to borrow Chemistry part Il, unless Chemistry part I is also borrowed, The number of ways in which 1s. he can choose the three books to be | jp \ | 19:-The interior angles of a regular polygon 18. 19. 2 points are joined and lines are formed, the maximum number of points of intersection that will form between the — lines is 1) 80 2/100 3)120 4) 240 measure 29” each. The number of diagonal of the polygon is Ns 29 3) 18 4)10 The number of rectangles excluding squares from a rectangle of size 9 x 6 is 1391 2)791~—«3)842— 4) 250 Ina plane there are two families of lines y=xtny=-x+n,wherer e {0,1,2,3, 4}. The number of squares of diagonals of the length 2 units formed by the lines is A rectangle with sides 2m -1, 2m - 1 is divided into squares of unit length by drawing parallel lines as shown in the diagram, ‘The number of rectangles with odd sidellength is D(m+n41y 2) mn(m+1) (n+l) 3) men? game Nine points lie in a plane forming a. as shown ee —— IfNis the number of triangles y area, having 1Je number of ways of selecting WO | «1 - squares from a chess board such that they neither have a common vertex nor have a ‘common side is N then last digit of N is 13 24 3)5 4)6 Three ladies have each brought their one child for admission to a school. The principal wants to interview the six persons one by one subject (o the condition that no mothers interviewed before her child. The number of ways in which interviews can be arranged is 16 2)36 3)72 4) 90. 24. In the next world cup of cricket there will be 12 teams, divided equally in two groups. ‘Teams of each group will play a’ match against each other. From each group 3 top teams will qualify for the next round. In this round each team will play against other once. Each of the four top teams of this round will play a match against the other three. Two top teams of this round will go to the final round, where they will play th best of three matches. The minimum number of matches in the next world cup will be 25. The (m,n), m,n €{1,2,...,100} number of ordered pairs such that 7" +7! is divisible by 5 is 2)2000 3)2500 4) 5000 1)1250 et number of ways of dividing 15 men and 15 women into 15 couples, each Y 4s consisting of a man and a woman is 1) 1240 2)1840 3) 15! 4) 2005 27. The number of ways of choosing 3 squares from a chess board so that they have exactly one common vertex 1)195 2) 196 3) 1974) 198 28. Number of squares of all dimensions of 5x7 game board TNO ie 92) 25 nu 3) 80) 4) 85 29. Arrange the following values in ascending [meee ee the given equation is ‘Arrange the following values * order. A: Nowof diagonals of a polygon es a No.of squares (exclusively squares) a chess board C:No.of waysin which 4 boys and six girly sit alternately in a row 1D: No.of sides of the polygon in which no.of sides is equal to no.of diagonals. 1) BADC 2)DCAB 3)CDBA 4)CDAB 31, The number of positive integral divisors of 1200 which are multiples of °6” is 32. If N is the least natural number which leaves remainders 2,4,6,10 when divided by 3,5,7,11 respectively, then the no. of visors of N is 4 = 25 3)6 47 33. The sum of positive integral divisors of 600 which are multiples of 10 is 1)840 2)3360 3) 1680 4) 420 34,The no. of ways can a group of 5 lettersbe formed out of Sa 's, 5b 's, Se 's and 5d 'sis 1)8C,x5 2)'C,x5 3)°C) Ayam In an election three districts are to be cam vassed by 2, 3 & 5 men respectively . If there are 10 men volunteer, the numberof Ways they can be alloted to the different 35. districts 10! 10! © rari @ ori 10! 10! @ => gn A2ty" 5! ® (27 3!St » 36 Let x.y.2=105 where x, y, ee number of ordered triplets (x © The number of positive integral coe StS OF the equation xyz =150 is 127 -2)54" ~~ 3)08 38. A test has 4 parts. The 10 marks each and the marks, Assuy tl inrecieea rae which aexamination of 9 papers a. candidate ‘has to pass in more papers than t of papers in which he fails in Ardiriste successful. The number of ways in which he can be unsuccessful is 1)285 2) 256 3) 193 4)319 ssi ots are five periodes in each working day ofa school, then the number of ways that you can arrange 3 subjects during the working day is 41. Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First the women choose the chairs from among the chairs marked I t 4, then the men select the chairs from among the remaining. The number of possible arrangements is 1) ‘P, . *P, 2) GPa BG. .“P: 4)4P, 4P, 42. 18 guests have to be seated half on each cular guests desire to sit on one particular side and 3 others on the other side. Then the number side of a long table. 4 pai of ways in which the sitting arrangements can be made ney 2) "Cc, 1? 3) "1, (91)? 4) "C1 Phere are 5 English, 4 Sanskrit and 3 Telugu books. Two books from each group are to be arranged in a shelf. The number of possible arrangements is 1) (180) 6! 2) (12) 7! 3)7! 4) 180 - The number of different combinations that can be formed out of the letters of the word ‘INFINITE! taken four at a time is ep 2228 3)28 4) 120 Ifthere are 5 periods in each working day ofa school, then the number of ways that you can arrange 4 subjects during the working day is 1)220 2)240 ~~ 3) 260 4) 280 ¢ number of different words which can be formed by taking 4 letters at a time out of the letters of the word 'EXPRESSION' is ee The number of permutations of the letters of the word 'INDEPENDENT' taken 5 at time is 1) 3302 2) 3320 45. 3) 3230 4) 3203 48. 8 . The number of ways in which a mixed |. The number of words of four ‘The number of ways that the le word "PERSON" can be pla squares of the adjoining figure: row remains empty R= R,- R,- 1) 20x6! 2) 26x6! 3) 20x5! 4) 26x5! How many different words can be formed by jumbling the letters in the word ‘MISSISSIPPI’ in which no two S are adjacent ? (AIEEE -2008) 1), 75GgaGe 2) 8°C, 7g SG er 4) 6.87C, Let ¢; 45 4be an invertible function where A= {1,2,3,4,5,6} The number of these functions in which at least three elements have self image is doubles tennis game can be arranged from: amongst 9 married couple if no husband and wife plays in the sam game is (2) 3024 (4) 6048 (1) 756 (3) 1512 . The number of permutations by using, the letter of the word MONDAY, such that which are not begining with M and ending with Y i . The number of positive integers < 10000 which contain exactly one 2, one S and on 7in its decimal representation is (1)2490 (2)2940 (3)2990 (4)2 containg equal number of yo consonants, repetition allowed is 1) 105? 2) 210x 3) 105x243 4) 150: Let n by any odd positive in greatest integer k such nent etPE i} " OTE RecN ues KEY 13 24 3)1 4)1 5)840 6)2 II 8)4 928 101 IL 12)1 13)2 14)226 15) 1 16)2 172 18)2 19)9 20)3 211 22)4 234 24) 53 25)3 26) 3 27)2 28)4 29)1 304 = 311232) 33)334)4 35) 1 36) 27 37)238)2—39)2 40) 150 411 42)4—43)1 44)2 45)2 46)2190 47)2 48) 2 49)1 50)56—51)3 52) 504 $3)2 54)255)9 HINTS Arational number of desired categoryis of the form 2002, XXpeeveek, Where |< k <9 and Spano >1 we can select K digits in °C, waysand arrange them in descending order in only one way. No. ofrational numbers is at +°C, 1 ()a,eP&a cQ2)acP&aeQ G)agP&aec ee € P&a, €Q A= {aa} a, e PAQ Bee satisfy (2) 3) 4) in ein) Ans*C, .3"? a en FOC) Paar Clo AG + BC+ PCy toner "Crp "C, (5c, ) (addand subtract "'C,, ) AG +E, + #C, +7 + “Gye Ca MC Bicwrc, Fy BC Cy tout Cao — Te dice 7, MC,-27 = 27378 8, From synopsis (He +(e, (AY hhdise contains the digits 0, 1, 2....9 Total no. of ways of dailing = 10° a Time taken to dial each disc = 5 sec si = Ars 60x60 x5 60x60 5 hrs of trial per 1 day hrs Total time taken = 10° > hrs of trial per how many days? 60x 60 2 7 fe SSUES © 0x60x5 7 =275, daysiie. 28 days 12;4,8,16; 5,10,20; 6,1 14,16; 1,5,25. The desi number is 6-+2-+4141=10 The total number of numbers without restrictions containing digits 2,3,4,5,6,1 n(S)=6" 7 The total number of numbers that 3,4,5,6,7 is n(A)=5" The total number of numbers that (8)=5". The total number of numbers that © 3,4,5,6isn(ANB)=4". 7 The total number of numbers tl contain digits 2 and 7 is 5" 45" number ofnumbers that 6-5" 5" 44". 12. Letaandbbe two ulEXERCISE -IIT SINGLE ANSWER QL STIONS ‘The number of numbers lying in (0, 1), whose all the digits after decimal are non zero and distinct is (A) DP, (B)9x 91) 101. (DY D'P, oi 7 There aren different white and n black balls marked 1, 2, 3, number of ways in which we can arra these balls in a row so that neighbouring balls are of different colours is different (2 (Ajo! (B)2n)!_ (C) 2 (n!)?(D) The number of all three digits even numbers such that if 3 is one of the digits, then next digit is (A) 359 (B) 360 How many four di by using the di repetition of a di; repetition of a di sis (C)365 (D) 380 it numbers can be made 1, 2, 3, 7, 8, 9 when it is not allowed respectively are (A)360,1296 (B)1296,360 (©300,400 (D)1200,300 (Ayo ®7 (©s (D9 Ten different letters of an alphabet are given. Words with five letters are formed from these of words which haye at least one letter repeated is (A) 69760 (B) 30240(C) 99748(D) 3 The streets of a city ged like the lines of a chess board . There are running North to South and ' in which a man can travel from NW to SE corner going the shortest possible distance is (A) fm? +n? (m+n (m+n-2)! (m-1)!.(n=1)! ‘The number oftimes the digit 3 will be written when listing the integers from I to 1000is © (D) m!.n! (A296 (B) 300 (C)271 (D) 302 10. 12. 13. 15, Let X)XpXyXy XoXo bea min palindrome such that either the sequene. (c:ctrety feets) isa strietly ascending or strictly descending. Then the number ofsuch palindromes is A)OXPP, B)3x°R, ©) 9%" Cy D) 3% C, An 8 digit number divisible by 9 is to be formed by using 8 digits out of the digits 9, 1,2,3, 4, 5,6, 7,89 without replacement, ‘The number of ways in which this can be done is (A)(34)7! (B)(35)7! (()(36)7! (D)(37)7! 7 boys be seated at a round table. X is the number of ways in which two particular boys are next to each other, Y is number of ways in which they are separated. then X+yY is (A)500 (B)600_ (C)700_ (D720 Number of ways in which 5 boys and 4 girls can be arranged on a circular table such that no two girls sit together and two particular boys are always together. (A)276 (BY: (D)304 Six persons A, B, C, D, E and Fare to be seated at a circular table. The numberof ways this can be done if A must have B or C on his right and B must have either CorD on his right is (A436 (B) 12. (©) 24 @) 18 Number of ways by which 4 letters canbe putin 4 correspondi elopes so that all letters go in wrong envelopes is (A)2 (BY (C44 (D265 The number of subsets of the set +, } which contains a even number of elements is (A) 2" (B) 2"_1 © 2-2) gt Rajdhani express going from Bombay Delhi stops at 5 intermediate stal 10 passengers enter the train durin journey with ten different tickets classes The number of different tickets they may have is (ANC, (BC, (CC, 1={a1,a9,.20. 2, . Given si 21, Te Rae e Uren ‘Number of ways of selecting 6 shoes, out ® Of 8 pairs of shoes, having exactly two pairs is (A) 1680 (B)240 (C) 120 (D) 3360 | The number of straight lines that can be drawn through any two points out of 10 points, of which 7 are collinear, (ays (B)30 (C35. (D) 45 ine segments of lengths 2, 3, 4, 5,6, 7units, the number of triangles that can be formed by these lines is WC-7 BC-6OC-s OC-4 . There are (p + q) different books on different topics in Mathematics, where p#q.IfL=The number of ways in which these books are distributed between two students X and Y such that X get p books and Y gets q books. M=The number of ways in which these books are distributed between two students X and Y such that one of them gets p books and another gets q books. N= The number of ways in which these books are divided into two groups of p books and q books, then (A) L=M=N (B) L=2M=2N (© 2L=M=2N (D) L=M=2N The sum of the divisors of 2°. 34. 5? is AZ. 7.10 2.7 17 31 (©)3.7. 11.31 31231 The number of numbers pq of the form 4n+1,where p,q, < {1,2,3, 5,7, 9 11} is (nez) (A)18 (B)36_- (C)42_- (D) 49 The number of positive integers that are the divisors of atleast one of the numbers eas’, 18!". is... (A) 1056 (B) 528 (C) 870 (D)435 The exponent of 2 in N=20x19*18%. -.%11. is (10 (BIS. (C20. (D2 The number of ways of distributing 8 identical balls in 3 distinct boxes so that of the boxes is is empty ws BBL OF 26. 27. 28. 29, 30. 31. 32. 33. . No. of different ways by which 3 p A, B and C having 6 one rupee one rupee coins and 8 one rupee 7 respectively can donate 10 one rupee coins collectively. (Ayal (B43 (C47 (DAD. ‘There are three piles of identical red, blue and green balls and each pile contains at least 10 balls. The number of ways of selecting 10 balls if twice as many red balls as green balls are to be selected, is (3 B84 (6 M8 The number of integral solutions for the ‘equation x + y +z +t =20, where x, y. Zt are all > —1,is (A)"C, (B)2C, (©) 7C, OVE; Number of positive integral solutions of 15 6) is Because Statement —2: There aren straight lines is a plane such that n, of them are parallel inone direction ,1n, are parallel in different direction and so on, n, are parallel another direction such that ny +n, +......+m% =n Ako no three of the given lines meet at a point Then the total number of points of intersection ;. Consider the word ‘SMALL’ Statement-1 : Total number of 3 letter words from the letters of the given word is 33. Statement—2 : Number of words having all the letters distinct = 24 and number of words having two letters alike and third different = 9 Statement—1: Number ofnon negative integral solutions of the equation x, +,#x,=10 is equal to 34. Statement-2: Number ofnon negative integral solutions of the equation X, +x, +x,+...+x, =r is equal to "*''C, Statement-1: 51 * 52 = 53 x 54 x 55 x 56 x 57 x 58 is divisible by 40320 Statement-2: The product ofr consecutive si numbers is always divisible by r! . The tamer of animals has to bring op, 39. 40. 41. 42. 43. MULTI ANSWER QUES by reus h this tiger, one five lions and four tigers to the ¢i arena. The number of ways in whic) can be done provided no two immediately follow each other is (A) 15. (B) 15 * 51 4! (©) 1800 (Dy 4359 You are given 8 balls of different coloy,, (black, white, ..). The number of ways, which these balls can be arranged in a roy, so that the two balls of particular coloy, (say red & white) may never come together is (A) 81-2.7! (B) 6.7! (©) 2.6!.C, (D) 2(7181) The maximum number of permutations of 2n letters in which there are only a's @ b's, taken all at a time is given by (A) 2C, 10 4n-6 4n So eernt n S Sle 2n-1 2n n+4 4 n-1 (2n-3)Qn ee es Number of permutations of the word " AUROBIND" in which vowels appear in the alphabetical order is (A) 'P, (B) (© 4!c, (D) °C, 5! The continued product, 2.6. 10.14 to n factors is equal to n! (A) *C, (B) ™P, (C) (n+ 1) (n+ 2) (n+ 3) (n+n) (D) °C, here are 3 sections in a question papel each containing 5 questions . A candidate has to solve only 5 questions, choosing atleast one question from each section « The number of ways in which he can make choices is (A)"C,-3"C,-3.5C, (B) 8C,3.'C,43.C, © 46, +3. 0, -3 40, ®) 18 (°C, . *C,+*C, .4C,)m7 46. 47. 48. 49, _ Awomen has 11 clo No.of different garlands that can ‘qhe kindergarden teacher has her class . She takes 5 of them ee ae to zoological garden as often as she cam, without taking the same kids more than once . Then the number of visits, the teacher a id rea to the garden exceeds that YY “s (B) ¥C, (©) *C, - *C, (D) #C, a e friends. Number of ways in which she can invite 5 of them to dinner, iftwo particular of them are not on speaking terms and will not attend together is (A) "'C,- °C (B) °C, © 3°C, (D) °C, Let N denote the greatest number of points in which m straight lines and neircles intersect. Then oa (Ay n/N-(” (B) m/N-("P,) (© N-("C2) iseven (D)N-"'Cy—"P, iseven Let n=" -p® -p’*..pfs where (P,,P, +P, are primes and a,, 0, a,-+- a, < N) then number of ways in which n can be expressed as the product of two positive factors which are prime to one another. Idup (Ao (B)2#_ (C2! ) 72 c The smallest number with 16 divisors is (A)15xsum of the coefficients in (1+ x) (B)2’ x3 (©)2?x3 (D)2’ x3! x5! Number of positive unequal integral Solutions of the equation x + y+ 7= 618 (A)41 (B)3!,- ()51(D)2*4! INTEGER QUESTIONS be made Using 5 flowers of one kind , 3 flowers of another kind is The number of ways in which 4 married can be sit four on each side of 8 52. 53. 54. 56. 57. 58. Passa; females on the other side and no wife is in_ front of her husband is k then k is ‘A guard of 12 men is formed from a group of ‘n° soldiers, It is found that 2 particular soldiers A and B are 3 times as often together on guard as 3 particulars soldiers C, D and E. Then(n-24) = ‘The number of triangles that can be formed with the angular points of a hexagon. Then the no. of triangles in which none of the sides are to be the sides of the hexagon The number of positive integral solution of the equation x,x,x,x,x, =1050 is 375m when 7c N.Then n= . Ifthe number of integral solutions of the 0 where equation — 2x+2y x>0,y20 and 220 is 11k then k= A seven digit number made up of alll distinet digits 8, 7, 6, 4, 2, x, y is divisible by 3. The possible number of order pairs (x. Consider S = {1,2,3,4,.....-,10}. Them sum ofall products of numbers by taking k two or more from S is (11! —k) then (| where [| is GI. Fis oo 1 1f4,=) ae then ‘Statement: A family group consisting of two men, three women and four children have fi row seats for a broad way show, where will all sit next to one another in seats through 109. The decision as to who si which seats must confirm to the folknro Ifa man is sitting in seat 107 and a woman. is sitting in seat 108, which of the following, could be the seat that another woman is sitting in? A)101 BY 102 C) 103.) 104 60. All three women will be sitting next to one another ifa man isin which of the following, seats? A) 104 B) 105) 107 D) 109 61, If women are sitting in seats 103 and 109, which of the following seats could be occupied by the third woman ? A) 102 B) 104 C) 106 D) 107 Ifa child is in seat 101 and a woman is in seat 105, which of the following pairs of seats must be occupied by people not of the same sex ? A) 10S and 107 B) 106 and 107 ) 106 and 108 D) 108 and 109 no Womanis sitting nextto another woman, | which of the following must be true? | A) Awoman is in seat 104 B) A childis in seat 105 | ‘C) Awoman is in seat 106 D) Amanisin seat 108 62. ‘Arrangement rounda circular table :A circular | table has no head and arrangements like these in the figures given below are considered identical Pot ra EXC, 8 ay7A Re E dD D Cc B A BE If persons are arranged in a straight line, there are n! different ways in which this can be done, When n persons sit round a circular table, each | circular arrangement will be equivalent to n arrangements in a line, so there are (n-1)! different arrangements of n persons round a circle. Alternatively, we can regard any one person as ‘head’ and place the other (n-1) person in (n-1)! different ways. A round robin conference of prime ministers of 40 countries, including India and Pakistan is to be held. ‘The nun ways in which prime ministers ean seated so that prime ministers of In Pakistan are never together, is (A) 37 «38! (B) 38 * 38! (C) 36 * 38! (D) 35 «38! ‘A round robin conference of prime ministers of 40 countries is to be held. The number of ways in which they can be seated such that prime ministers of ‘America and Britain are always together and those of Russia and India are always 65. together, is (A)2* 37! (C2 38! (B)4* 37! j (D) 4 «38! In the above question, if prime ministers of India and Pakistan do not sit together, then number of ways is (A) 36 x 36! (B) 72 «36! (© 144 «36! (D) 288 = 36! . A round robin conference of prime ministers of 40 countries is to be held. The number of ways in which they cam be seated so that prime minister of India neither sits with Pakistani counterpart nor with Chinese counterpart, is (A) 111 x 37! (B) 333 x37! (C) 666 x 37! (D) 1332 x 37! 66. Let x, x,x, X, X,x, be asix digit number find the number of such numbers % SX, » divides P, then the maximum value of x is A)45 B)90 ~——-C) 135 D) 180 If the sum of the reciprocals of all the positive integral divisors of N is R then 2 B. (D) 451024 4. 75. 16. [2] is (where [R] is the greatest integer less than or equal to R) A)l B)2 Passage — VI Two numbers x and y are drawn without replacement from the set of the first 15 natural numbers. The number of ways of drawing them such that C)3 D)4 7, x' +,° is divisible by 3 A)21 B)33. ©) 35D) 69 is divisible by 5 B)33 x‘ — y' is divisible by 5 A)57 B) 64 Passage - Passage - VII Five balls are to be placed in three boxes. Each can hold all the five balls. The no. of ways of Placing the balls so that no box remains empty. 78. ©) 35 D) 69 0, ©) 69 D) 72 ©. ifballs and boxes are all different is (A)so — (B) 100 (C) 125 (D) 150 81. if balls are identical but! is a (A)2 (B)6 (C4 82. Ifhallsare different but boxes: (A)25. (B)1S. (C)10 83. if balls as well as boxes are (A)1 (B)2 — (C)25 ass 1 Ifa set A has ‘n’ elements then the nun subsets of A containing exactly “r” is” C,, The number ofallsubsetsofAis 2%. N answer the following questions. Aset Ahas7” clements. Asubset Pof Ais selected. A fternoting the elements they are placed back in A. Againsubset is selected Then the number of ways of select and Qsuch that . P,Q have no common element is A) 2835 B) 128 C) 3432 D) 2187 . PandQ have exactly 3 elementsin common is A) 2835 B) 128 C) 3432 D) 2187 Pand Q have equal number of elements is (P and Q may be null sets) A) 2835 B) 128 C) 3432 D) 2187 MATRIX MATCHING QUESTIONS: 87. Match the following: Consider all possible permutations of the letters of the word MASTERBLASTERS & ‘Column -1 . A) The number of permutations containing the r, word RAAT is B) The number of permutations in which $ _ occurs in first place and R occurs im the last place is C) The number of permutations in whieh ofthe letters 8, T, R occurin first 7p letters A, S, R occur in even positions is Column - I f (7)? Lika! P)312y' © 3x (2y° ® 12!3 12. . Ani . An unlimited number of coupons bearing Four distinct bans B Dydd are to be Placed in 5 distinct boxes By By By By Be not more than one in a box. The numer of ‘ways in which they can be arranged so that 4 will not go to 8, for any “4” is (D(n) denotes de-arrangment) A) D(s) B) SD(4) ©) D(s)+D(4) D) D(5)+5D(4) Let 4={1,2,3.......n}. The no. of bijections from onto A forwhich (1) #1 is A)ni-n On? B)n!-(n-1)! n D)(n-1)! If C, denotes *C, , then the value of 2 A)12 re a +x) ® B) 13 Cc) 14 D)IS cream parlour has ice creams in eight different varieties. Number of ways of choosing 3 ice creams taking atleast two ice creams of the same variety, is : A) 56 B) 64 C) 100 D)27 (Assume that ice creams of the same variety are identical & ayailable in unlimited supply) the letters A, Band C are available, then the number of ways of choosing 10 of these coupons so that they can’t used to spell BAC A) 3(2”-1) B) 2(3"°-1) ©) 2-1 D) 2" Using the points from 4x5 an array of (A)28 (BY 112. (C) Hat Tad pibduetof al ba Ife diye Pg tl mum value of x is (A)28 (B)30.— (C32 Number of zeros at the end is Ayo Bl C2 . The sum of those factors of 7! w of form 3n + 1 and odd (1 4) B = {¥yy Yas Yad the total number of functions f: A > B. that are onto and there are exactly four element (x) in Asuch that f(x) =y,, is equal to (A)14 *C, (B)16 x 8G, (C)l4 x, (D)16 x *C, ‘The number of ways of dividing 15 n and 15 women into 15 couples (each e* consists of a man and a woman) wh order of the couples is conside: AD? 27 BS38. 39. 40. 41. 42. - In how many wa: sticks of length 1 cm each are distributed into three children A,B and C. ‘These children join the sticks in the form of line segments individually. If ‘n’ is the in which the sticks can be the children so that the line segments joined by them form a triangle A) Number of distributions so that they form triangle = 10 B) Number of distributions so Tight angle triangle = 3 ©) Number of distributions so that they form isossceles but not equilateral =0 D) Number of distributions so that they form isossceles right angle triangle =0 Ibe the vertices polygon inscribed in acircle with centre O. Triangles are formed by joining the vertices of the 21-sided polygon (A) The number of equilateral triangles is 7 (B) The number of isosceles triangles is 196 at they form (D) The number of isosceles triangle is 186 INTEGER QUESTIONS 5 different rings can be arranged in four fingers. (One finger can hold all the five rings). Number of positive terms in the sequence 16 mal 4P. The number of thre iddle digi digits is nen, P it numbers, whose is bigger than the extreme If n, and 7, digit no.s, find the total no.of ways of for ing n, &n, so that these numbers can be added without carrying at any stage is 36(55)' then the value of k The no.of +ve integers from 1 to 1000, which are divisible by at least 2,3 or 5 is A father has 5 pair wise distinct oranges, xis the number of ways he gives away all the oranges to his 8 sons such that each gets one ornone. y isthe number of ways he gives away all the oranges to his 8 sons 43. 44. 46. 47. 48, 49, 50. . The number of de-arrangements of ‘when each can get any number of y tn) ‘he number of distinct natural nump, ae maximum of 4 digits and clan by 5, which can be formed with theqigie 0, 1, 2, 3, 4 5, 6, 7, 8, 9 each digit pp occuring more than once in each nung. earof 100 people at a round path3 to be choosen the number of Ways 50 thy no two off the choosen consecutive jy 100 96 © | then the value of kei {1,2,3,4,5,6} to {1,2.3,4,5,6} such that p mapped only “2 is 6! 1 xe ili value of K is 1 1 ies 7 | then the A set contains (27 +1) elements, If the number of subsets of this set which contains atmost ‘n’ elements is 4096 then the value of ‘n? Four persons A, B,C,D are to be seatedin a row such that B does not followA, C does not follow B and D does not follow C. Then the number of ways of seating themis/ 1 then =____(/=GLR) 10 Number of quadrilaterals that can be made using the vertices of a polygon of 10 sides having 3 sides common with the polygon is (K+3), then k Number of triangles that ean be made using the vertices of a polygon of 10 their vertices and having exactly common with the polygon is 10k, There Were ‘n’ circles in a bot maximum number of regions equal to K +25) (n—1), then Kisjer vertices of cube, The number of id Gomera triangles can be made using the vertical number N has exaetly 12 2 tinct (positive) divisors including itseit ind 1, Duc only 3 distinct prime factors, 1f ant um of these prime factors ig 29, the pute te smallest possible value of Ny ghim ofthe digits oF Cotal number of ways ss Selecting § letters from the letters of tye word INDEPENDENT, jnanexamination, (he max mark for exch ofthe 3 papers is SO and the max mark for the fourth paperis 100. The no.of ways in which the cand date can score 60% marks in aggregrate is ‘The number of integers which lie between tand 10° and which have the sum of the digits equal to 12. 4 Thenumber of ordered triplets of positive ™ integers which satisfy the inequation °C, then kis sj, Number of pairs of positive integers (p,q) whose LCM (Least common multiple) is 8100, is “K”. Then number of ways of expressing K as a product of two co-prime numbers is 58, How many even no.s are there with 3 digits such that the next digit? COMPREHENSION QUESTIONS & sx+y+z<50 is “C, PASSAGE - sD, y,, are 1000 doors “and, P,, re 1000 persons. Initially allthe doors are closed. P, opensall the doors. Then, P, closes D,,D, D,— Dyygy Dyooo Then P, changes the status of D, ,D, D, Dys— ete, (doors having numbers which are multiples of 3). Changing the status ofa door Means closing it if it is open and opening it ifit closed, Then P, changes the status of D,,D, 12D, —ete (doors having numbers which ate multiples of 4), And so on until lastly P 49) % es the status Of D ssp: * Finally, how many doors are open? @ 029 B31 c)32_——«D)33 hat is the greatest number of 61. ‘The door having the is finally open is A) Consider the letters of ‘MATHEMATICS’ Possible number of words taking. time such that at least one repe: letter is at odd position in each word 62, ut 9! 9 22101 2101 =) Za 9 ut O70 D) ayaa 63. Possible number of words taking all letters at a time such that in each word both “M’s are together and both “T’ s are together but both ‘A’s are not together is il! 10! Fan 2a «B74 » ia a 212! >) 300108 64. Possible number of words in which no two. vowels are together is 3c 4 Ae 4! 1 4 oy D) da at PASSAGE -III: Consider the network ofequally spaced parallel lines (6 horizontaland9 vertical) shown in the figure. All small squares are ofthe same size. A shortest route from A to C is defined asa route consisting 8 horizontal steps: and 5 vertical steps. Since any shortest route: is a typical arrangement of 8H and SV, The ! zs 135 number of shortest routes SLs . Answer °C ae 212! the following questions F . The number of shortest routes through: the junction P..... secutive doors that are closed finally? )56 B)sg = c)60~—«i@DY 62 A)240-B)216 ©)Grier Trent 66. The number of shortest routes which go following street PQ must be... A)324 BB) 350. -C)512._—«(D). 256 67. The number of shortest routes which pass through junctions P and R... A)l44_B)240 C)216 ~——D) 256 Vv: Let @=(a,,a,,4,,....,a,) be a given arrangement of n distinct objects 4,944 4,.....54,. A derangement of is an arrangement of these n objects in which none ofthe objects occupies its original position. Let D, be the number of derangements of the permutation @ 68. D, is equal to A) (n=1)D,,+D,, B) D,,+(n-1)D,, ©)n(D,.,+D,2) D)(n-1)(D,,+D,.) 69. The relation between D, and D, , is given by A) D,—nD,_, = B) D, C) D, -nbD,, Z D) D, -D,.,=(-1)"' 70. There are 5 different colour balls and 5 boxes of colours same as those of the balls. The number of ways in which one can place the balls into the boxes, one each in a box, so that no ball goes to a box of its own colour is A)40—-B)44 ©)45 D)60 PASSAGE -V : Let A={I,2,3,.....7} bea set containing nelements 71. Forany given three number of subsets of A haying k (k ie mambers are formed using the digits 4,234.5. Match the statements in Column-I with the josgiven in Column-II, )LUMN - I {Ay How many of them are divisible by 3 if nis not allowed B) How many of them are divisible by 3 if petition of digits is allowed ()How many of them are divisible by 3 but totby2 ifrepetition is not allowed D)Number of 4 digit numbers divisible by 5 {without repetition) COLUI i P26 Q) 108 ~—-R) 2160 S) 42 BA1I7 member hockey squad contain 4 Particular players A,B,C and D. Players A and B wish to play together or be out of the team together. Players C and D are Such that if one plays the other does not Want to play. A team of all players is to be Selected from the squad. the items in Column-I with those in ll, COLUMN. ' rpotieectons including A and B and one gboth Cand D is N°, of selections excluding A and B and wg one of C and D is ©. Of selections excluding all of A,B,C & 85. children each 18, is 8) The number of ways of forming one having 5 numbers choosen from 5 boys and 5 girls, so that girls are in majority and atleast ‘One boy is there in the team. ; ©) Six bundles of books are to be kept im boxes one in each box. If 2 of the Boxes: too small for three of the bundles, then of ways keeping the bundles in the boxes i D) Abag contains 6 black, 6 blue, 6 red, 6; ineach colour. The number of ways of 2 balls from the bag such thatthe balls same colour or the numbers on them: s COLUMN - II P) 125 Q 127 R) 135 S) 144 4 Match the statements in Column-I with 2 points ina plane out 5 are collinear and no 3 of the remaii B) Theno.ofangles that can bet the points contained above is. C) Theno.of rectangles that cau using the squares ina chess! D) Ifaset of 8 parallel lr another set of 6 parallel linMatch the statements in Column-L with the Nos given in Column-II. A) ‘Number of ways of distributing 7 books to 2 children is B)The value of 74 C44 Get, tC, ©) The number of five digit numbers in which cvery digit exceeds the immediately preceding digit is D) Number of permutations of {1,2,3,4,5,6} Such that the pattems 13 or 246 do not appear COLUMN-11 P) 126 Q 128 R)582 $)220 SINGLE ANSWER QUEST: IONS Ic 2D 3)D 4B Bik 6)B DG) Bete) De TON) MA 12)C 13)B 14)B 15)B 16)D INA 18)A 19)B20)C 21I)A 22)C 23)C_24)A_ 25) ASSERTION-REASON QUESTIONS 26.C MULTIANSWER QUESTIONS eB) 28)AB 29)C 30) A 32).B,C 33)A 34) CD INTEGER QUESTIONS 38.4 Bi 39,240. 40.4 41. 734 42.4 43.1106 44.3 45.5 46.6 47.1 48.7 49.6 50.2 518 52.260 53.9 54.110551 55.6062 56.3 57.2) 58,365 COMPREHENSION \ QUESTIONS 59.B 60.C 62.D 63.B 64.C 6: ¢ 66.B 67.B 68.D Ba ic RC 7.C 14.D ISA 76.A 77.0 BA 79D 80.A 8LA MATRIX MATCHING 82. (A-P), (B-R), (C-Q), (D-S) 83. (AR. (BP (C-S),(D-Q) 84. (A-Q), (B-P), (C-S), (D-R) B.S) (C2). DE) 86. (A-Q), (B-S), (C-P), (D-R) 10, lh, HINTS — SINGLE ANSWER Q Let anarranngement be 24, Product of any five consecutive: by 7. This is possible only when y that total number of ways is gy Use the concept (total no.of per permutations not repeated) X(*) = sumofall 3 digited natu i formed by using 1, 3, 5,7,9 =25x(1+3+5+7+9) x(II1) 9.375 Because each digit occur in each of unig, 10’ place and 100’s place (55) b — Suppose the arrang XXXX,XX,X,X,. Lf we select five es positions both x, and x, will always be: and either of them should contain, Hence 2° 7! = 3331 x 3337 Note : N— 9 = (3334)2_ 32 = (3337) x (3331)P N —9 cannot bey UL Nollissoscoeaay £33 =p nl=k a= k=12 2 Consider dummy ball b, and applied — de-arrangment total no.of arrangements - a which f( (0 +2?43°442)<15 i cua 56 = 64 : When all the selected co letter. One letter can| three letters in %e, ways => Ways of choosing 10 co