PERMUTATIONS AND COMBINATIONS

www.sakshieducatio.com PERMUTATIONS AND COMBINATIONS OBJECTIVE PROBLEMS. There are parcels ad 5 post-offices. I how may differet ways the registratio of parcel ca be made 5 (a) 0 (b) 5 (c) 5 (d) 5. I how may ways books ca be arraged i a row so that two specified books are ot together (a)! ( )! (b) ( )!( ) (c)! ( ) (d) ( )!. I how may ways ca 0 true-false questios be replied (a) 0 (b) 00 (c)5 (d)0. There are 5 roads leadig to a tow from a village. The umber of differet ways i which a villager ca go to the tow ad retur back, is (a) 5 (b) 0 (c) 0 (d) 5 5. Six idetical cois are arraged i a row. The umber of ways i which the umber of tails is equal to the umber of heads is (a) 0 (b) 9 (c) 0 (d) 0. Assumig that o two cosecutive digits are same, the umber of digit umbers, is (a)! (b) 9! (c) 9 (d) 9 7. The sum of all digit umbers that ca be formed by usig the digits,,, 8 (repetitio of digits ot allowed) is (a) 0 (b) 580 (c) 58 (d) Noe of these www.sakshieducatio.com

www.sakshieducatio.com 8. The umber of umbers that ca be formed with the help of the digits,,,,,, so that odd digits always occupy odd places, is (a) (b) 8 (c) (d) 0 9. The umber of words which ca be formed from the letters of the word MAXIMUM, if two cosoats caot occur together, is (a)! (b)!! (c) 7! 0. The value of P r is equal to (d) Noe of these (a) P r + r Pr (b). P + r Pr (c) ( Pr + Pr ) (d) Pr + Pr. Numbers greater tha 000 but ot greater tha 000 which ca be formed with the digits 0,,,, (repetitio of digits is allowed), are (a) 50 (b) 75 (c) 50 (d) 57. I how may ways ca 0 balls be divided betwee two boys, oe receivig two ad the other eight balls (a) 5 (b) 75 (c) 90 (d) Noe of these. There are parcels ad 5 post-offices. I how may differet ways the registratio of parcel ca be made 5 (a) 0 (b) 5 (c) 5 (d) 5. How may words ca be made from the letters of the word COMMITTEE 9! (a) (!) (c) 9!! 9! (b) (!) (d) 9! 5. I how may ways ca 5 boys ad girls sit i a row so that o two girls are together (a) 5!! (b) P 5! 5 (c) P 5! (d) P! www.sakshieducatio.com

www.sakshieducatio.com. All the letters of the word EAMCET are arraged i all possible ways. The umber of such arragemets i which two vowels are ot adjacet to each other is (a) 0 (b) (c) 7 (d) 5 7. How may umbers ca be made with the digits,, 5,, 7, 8 lyig betwee 000 ad 000 which are divisible by 5 while repetitio of ay digit is ot allowed i ay umber (a) 0 (b) (c) 0 (d) 8. How may umbers cosistig of 5 digits ca be formed i which the digits, ad 7 are used oly oce ad the digit 5 is used twice (a) 0 (b) 0 (c) 5 (d) 90 9. I how may ways letters ca be posted i letter-boxes, if all the letters are ot posted i the same letter-box (a) (b) 0 (c) 77 (d) 8 0. The total umber of permutatios of the letters of the word BANANA is (a) 0 (b) 0 (c) 70 (d). How may words ca be formed with the letters of the word MATHEMATICS by rearragig them (a) (c)!!!!!!! (b)!! (d)!. How may umbers less tha 000 ca be made from the digits,,,, 5, (repetitio is ot allowed) (a) 5 (b) 0 (c) 50 (d) Noe of these. How may umbers greater tha hudred ad divisible by 5 ca be made from the digits,, 5,, if o digit is repeated (a) (b) (c) (d) 0 www.sakshieducatio.com

www.sakshieducatio.com. If a deotes the umber of permutatios of x + thigs take all at a time, b the umber of permutatios of x thigs take at a time ad c the umber of permutatios of x thigs take all at a time such that (a) 5 (b) (c) 0 (d) 8 a 8 bc, the the value of x is 5. The product of ay r cosecutive atural umbers is always divisible by (a) r! (b) r (c) r (d) Noe of these. The umber of ways i which te cadidates A, A,... A0 ca be raked such that A is always above A 0 is (a) 5! (b) (5!) (c) 0! (d) (0!) 7. How may umbers lyig betwee 999 ad 0000 ca be formed with the help of the digit 0,,,,7,8 whe the digits are ot to be repeated (a) 00 (b) 00 (c) 00 (d) 00 8. The umber of digit eve umbers that ca be formed usig 0,,,,, 5, without repetitio is (a) 0 (b) 00 (c) 0 (d) 0 9. The umber of words that ca be formed out of the letters of the word ARTICLE so that the vowels occupy eve places is (a) (b) 57 (c) (d) 75 0. If the letters of the word SACHIN arraged i all possible ways ad these words are writte out as i dictioary, the the word SACHIN appears at serial umber (a) 0 (b) 0 (c) 0 (d) 00 www.sakshieducatio.com

www.sakshieducatio.com. The umber of arragemets of the letters of the word BANANA i which two N s do ot appear adjacetly is (a) 0 (b) 0 (c) 80 (d) 00. If a ma ad his wife eter i a bus, i which five seats are vacat, the the umber of differet ways i which they ca be seated is (a) (b) 5 (c) 0 (d) 0. Total umber of four digit odd umbers that ca be formed usig 0,,,, 5, 7 are (a) (b) 75 (c) 00 (d) 70. Let the eleve letters A, B...,K deote a arbitrary permutatio of the itegers (,,...), the ( A )( B )( C )...( K ) (a) Necessarily zero (c) Always eve (b) Always odd (d) Noe of these 5. The umber of ways i which me ad 5 wome ca die at a roud table if o two wome are to sit together is give by (a)! 5! (b) 0 (c) 5!! (d) 7! 5!. If eleve members of a committee sit at a roud table so that the Presidet ad Secretary always sit together, the the umber of arragemets is (a) 0! (b) 0! (c) 9! (d) Noe of these 7. The umber of ways i which 5 beads of differet colours form a ecklace is (a) (b) (c) 0 (d) 0 8. I how may ways 7 me ad 7 wome ca be seated aroud a roud table such that o two wome ca sit together (a) ( 7!) (b) 7!! (c) (!) (d) 7! www.sakshieducatio.com

www.sakshieducatio.com 9. I how may ways a garlad ca be made from exactly 0 flowers (a) 0! (b) 9! 9! (c) (9!) (d) 0. I how may ways ca 5 boys ad 5 girls sit i a circle so that o two boys sit together (a)! 5! 5 (b)! 5! (c) 5! 5! (d) Noe of these. The umber of ways that 8 beads of differet colours be strig as a ecklace is (a) 50 (b) 880 (c) 500 (d) 0. I how may ways ca getleme sit aroud a roud table so that three specified getleme are always together (a) 9! (b) 0! (c)!0! (d)!9!. The umber of ways i which 5 male ad female members of a committee ca be seated aroud a roud table so that the two female are ot seated together is (a) 80 (b) 00 (c) 70 (d) 80. C + r C r is equal to (a) + C r (b) C r+ + (c) C r+ (d) C r 5. A ma has 7 frieds. I how may ways he ca ivite oe or more of them for a tea party (a) 8 (b) 5 (c) 7 (d) 0. If, C 8 C ad r r C r+, the the value of r is (a) (b) (c) (d) Noe of these 7. Everybody i a room shakes had with everybody else. The total umber of had shakes is. The total umber of persos i the room is (a) (b) (c) (d) www.sakshieducatio.com

www.sakshieducatio.com 8. If is eve ad the value of C r is maximum, the r (a) (c) (b) + (d) Noe of these 9. If C : C :, the for which of the followig values of r, the value of C will be 5 r (a) r (b) r (c) r (d) r 5 5 5 50. If C r C r+, the the value of r is (a) (b) (c) 5 (d) 8 5. I a electio there are 8 cadidates, out of which 5 are to be choose. If a voter may vote for ay umber of cadidates but ot greater tha the umber to be choose, the i how may ways ca a voter vote (a) (b) (c) 8 (d) Noe of these 5. I a city o two persos have idetical set of teeth ad there is o perso without a tooth. Also o perso has more tha teeth. If we disregard the shape ad size of tooth ad cosider oly the positioig of the teeth, the the maximum populatio of the city is (a) (b) () (c) (d) 5. How may words ca be formed by takig cosoats ad vowels out of 5 cosoats ad vowels (a) 5 C C (b) 5 C C 5 (c) 5 C C (d) ( 5 C C ) (5)! 5. There are 5 persos i a party ad each perso shake had with aother, the total umber of hadshakes is (a) 5 P (b) 5 C (c)! 5 (d) (5!) www.sakshieducatio.com

www.sakshieducatio.com 55. There are 5 persos i a party ad each perso shake had with aother, the total umber of hadshakes is (a) 5 P (b) 5 C (c)! 5 (d) (5!) 5. If 8, C C ad r r C r +, the equals (a) 8 (b) 9 (c) 0 (d) 5 57. I a electio the umber of cadidates is greater tha the persos to be elected. If a voter ca vote i 5 ways, the the umber of cadidates is (a) 7 (b) 0 (c) 8 (d) + 58. If C C, the (a) (b) (c) 5 (d) 59. I a electio there are 5 cadidates ad three vacacies. A voter ca vote maximum to three cadidates, the i how may ways ca he vote (a) 5 (b) 0 (c) 0 (d) 5 0. Six + ad four sigs are to be placed i a straight lie so that o two sigs come together, the the total umber of ways are (a) 5 (b) 8 (c) 5 (d). The umber of ways of dividig 5 cards amogst four players equally, are 5! (a) (!) (b) 5! (!)! (c) (!) 5! (d) Noe of these (!). Out of 0 white, 9 black ad 7 red balls, the umber of ways i which selectio of oe or more balls ca be made, is (a) 88 (b) 89 (c) 879 (d) 89 www.sakshieducatio.com

www.sakshieducatio.com. A total umber of words which ca be formed out of the letters a, b, c, d, e, f take together such that each word cotais at least oe vowel, is (a) 7 (b) 8 (c) 9 (d) Noe of these. Out of books, i how may ways ca a set of oe or more books be chose (a) (b) (c) (d) 5 5. All possible two factors products are formed from umbers,,,,..., 00. The umber of factors out of the total obtaied which are multiples of 5 is (a) 500 (b) 780 (c) 850 (d) Noe of these. The umbers of permutatios of thigs take r at a time, whe p thigs are always icluded, is (a) p! (b) p r! C r p (c) C r! (d) Noe of these r p C r 7. I a tourig cricket team there are players i all icludig 5 bowlers ad wicketkeepers. How may teams of players from these, ca be chose, so as to iclude three bowlers ad oe wicket-keeper (a) 50 (b) 70 (c) 750 (d) 800 50 5 8. The value of C + r C is r (a) 5 C (b) 5 C (c) 55 C (d) 55 C 9. A studet is to aswer 0 out of questios i a examiatio such that he must choose at least from the first five questio. The umber of choices available to him is (a) 0 (b) 9 (c) 80 (d) www.sakshieducatio.com

www.sakshieducatio.com 70. If C r deotes the umber of combiatios of thigs take r at a time, the the expressio r+ + C r + C r equals C (a) + + Cr (b) r+ C (c) + C (d) + r C r+ 7. A father with 8 childre takes them at a time to the Zoological gardes, as ofte as he ca without takig the same childre together more tha oce. The umber of times he will go to the garde is (a) (b) (c) 5 7. C r ( k ). Cr+ if k (d) Noe of these (a) [, ] (b) (, ) (c), ) ( (d) (, ) 7. The umber of ways i which thirty five apples ca be distributed amog boys so that each ca have ay umber of apples, is (a) (b) (c) (d) Noe of these 7. A perso is permitted to select at least oe ad at most cois from a collectio of ( + ) distict cois. If the total umber of ways i which he ca select cois is 55, the equals a) (b) 8 (c) (d) 75. The umber of ways i which four letters of the word MATHEMATICS ca be arraged is give by (a) (b) 9 (c) 80 (d) 5 7. I how may ways ca 5 red ad white balls be draw from a bag cotaiig 0 red ad 8 white balls 8 0 0 8 (a) C 5 C (b) C 5 C (c) 8 C 9 (d) Noe of these www.sakshieducatio.com

www.sakshieducatio.com 77. A studet is allowed to select at most books from a collectio of ( + ) books. If the total umber of ways i which he ca select oe book is, the the value of is (a) (b) (c) (d) Noe of these 78. Let T deote the umber of triagles which ca be formed usig the vertices of a regular polygo of sides. If T T, the equals (a) 5 (b) 7 (c) (d) + 79. The straight lies I, I, I are parallel ad lie i the same plae. A total umber of m poits are take o I, formed with vertices at these poits are m (a) k C poits o I, k poits o I. The maximum umber of triagles C C C C + + (b) m+ + k m k m k (c) C C C + + (d) Noe of these 80. If a polygo has diagoals, the the umber of its sides are (a) 7 (b) (c) 8 d) Noe of these 8. The umber of triagles that ca be formed by choosig the vertices from a set of poits, seve of which lie o the same straight lie, is (a) 85 (b) 75 (c) 5 (d) 05 8. There are m poits o a straight lie AB ad poits o aother lie AC, oe of them beig the poit A. Triagles are formed from these poits as vertices whe (i) A is excluded (ii) A is icluded. The the ratio of the umber of triagles i the two cases is (a) (c) m + m + m + m + + (b) m + (d) Noe of these 8. The greatest possible umber of poits of itersectio of 8 straight lies ad circles is (a) (b) (c) 7 (d) 0 www.sakshieducatio.com

www.sakshieducatio.com 8. There are straight lies i a plae, o two of which are parallel ad o three pass through the same poit. Their poits of itersectio are joied. The the umber of fresh lies thus obtaied is (a) ( )( ) 8 (b) ( )( )( ) (c) ( )( )( ) 8 (d) Noe of these 85. A parallelogram is cut by two sets of m lies parallel to its sides. The umber of parallelograms thus formed is (a) ( C ) m m + (b) ( ) C m + (c) ( ) C (d) Noe of these 8. Out of 8 poits i a plae, o three are i the same straight lie except five poits which are colliear. The umber of (i) straight lies, (ii) triagles which ca be formed by joiig them is (i) (a) 0 (b) (c) (d) (ii) (a) 8 (b) 80 (c) 800 (d) 750 87. The umber of parallelograms that ca be formed from a set of four parallel lies itersectig aother set of three parallel lies is (a) (b) 8 (c) (d) 9 88. Out of 0 poits i a plae are i a straight lie. The umber of triagles formed by joiig these poits are (a) 00 (b) 50 (c) 0 (d) Noe of these 89. Out of 0 poits i a plae are i a straight lie. The umber of triagles formed by joiig these poits are (a) 00 (b) 50 (c) 0 (d) Noe of these 90. Give six lie segmets of legths,,, 5,, 7 uits, the umber of triagles that ca be formed by these lies is (a) C 7 (b) C (c) C 5 (d) C www.sakshieducatio.com

www.sakshieducatio.com 9. There is a rectagular sheet of dimesio ( ) m ( ), (where m > 0, > 0). It has bee divided ito square of uit area by drawig lies perpedicular to the sides. Fid umber of rectagles havig sides of odd uit legth m (a) ( m + + ) (b) m ( m + )( + ) m + (c) (d) m 9. The sum m 0 0, i 0 i m i where p 0if p < q, is maximum whe m is q (a) 5 (b) 5 (c) 0 (d) 0 9. If P 80, C 5, the is equal to r r (a) (b) (c) 5 (d) 7 9. The umber of way to sit me ad wome i a bus such that total umber of sitted me ad wome o each side is (a) 5! (b) C 5! 5 (c)! P5 (d) 5! + C 5 95. A -digit umber is a positive umber with exactly digits. Nie hudred distict - digit umbers are to be formed usig oly the three digits, 5 ad 7. The smallest value of for which this is possible is (a) (b) 7 (c) 8 (d) 9 9. Number of ways of selectio of 8 letters from letters of which 8 are a, 8 are b ad the rest ulike, is give by 7 8 (a) (b) 8. 7 (c) 0. (d) Noe of these www.sakshieducatio.com

www.sakshieducatio.com 97. A set cotais ( + ) elemets. The umber of sub-sets of the set which cotai at most elemets is (a) (b) + (c) (d) 98. The umber of umbers of digits which are ot divisible by 5 are (a) 700 (b) 00 (c) 00 (d) 800 99. The umber of ordered triplets of positive itegers which are solutios of the equatio x + y + z 00 is (a) 005 (b) 85 (c) 508 (d) Noe of these 00. The umber of divisors of 900 icludig ad 900 are (a) 0 (b) 58 (c) 8 (d) PERMUTATIONS AND COMBINATIONS HINTS AND SOLUTIONS. (c) Required umber of ways are 5.. (b) Total umber of arragemets of books!. If two specified books always together the umber of ways ( )! Hece required umber of ways! ( )! ( )! ( ) ( )! ( ).. (d) Required umber of ways are 0 0, because every questio may be aswered i ways.. (a) The ma ca go i 5 ways ad he ca retur i 5 ways. Hece, total umber of ways are 5 5 5.! 70.!! 5. (a) Required umber of ways 0. (a) cocept www.sakshieducatio.com

www.sakshieducatio.com 7. (a) Sum of the digits i the uit place is ( + + + 8) 0 uits. Similarly, sum of digits i te place is 0 tes ad i hudredth place is 0 hudreds etc. Sum of all the umbers is 0( + 0 + 0 + 0 ) 0 0. 8. (b) The odd digits,,, ca be arraged i the odd places i ways ad eve!!!! digits,, ca be arraged i the three eve places i ways. Hece the required! umber of ways 8 9. (a) Cocept 0. (a) C r + Cr Cr. Pr r! + Pr ( r )! Pr r! r + r. Pr Pr. P. (b) Numbers greater tha 000 ad less tha or equal to 000 will be of digits ad will have either (except 000) or or i the first place with 0 i each of remaiig places. After fixig st place, the secod place ca be filled by ay of the 5 umbers. Similarly third place ca be filled up i 5 ways ad th place ca be filled up i 5 ways. Thus there will be 5 5 5 5 ways i which will be i first place but this iclude 000 also hece there will be umbers havig i the first place. Similarly 5 for each or. Oe umber will be i which i the first place ad i.e. 000. Hece the required umbers are 5 + 5 + 75 + ways. 0!! 8!. (c) A gets, B gets 8; 5 0! 8!! A gets 8, B gets ; 5 5 + 5 90.. (c) Required umber of ways are 5.. (b) CONCEPT 5. (c) Sice the 5 boys ca sit i 5! ways. I this case there are places are vacat i which the girls ca sit i P ways. Therefore required umber of ways are 5! P.. (c) First, we arrage cosoats i! ways ad the at four places (two places betwee them ad two places o two sides) vowels ca be placed i! P ways. P.! Hece the required umber! 7 www.sakshieducatio.com

www.sakshieducatio.com 7. (b) must be at thousad place ad sice the umber should be divisible by 5, so 5 must be at uit place. Now we have to filled two place (te ad hudred) i.e., 5! 8. (b) Required umber of ways are 0.! P. 9. (b) Three letters ca be posted i letter boxes i ways but it cosists the ways that all letters may be posted i same box. Hece required ways 0.!.!! 0. (a) Total o. of permutatios 0. (c) Sice there are M's, A's ad T's. Required umber of ways are!.!!!. (a) Number of digit umbers P Number of digit umbers P Number of digit umbers P The required umber of umbers + 0 + 0 5. (b) Stadard problem. (b) We have x+ x a Px + ( x )!, b P. x! + ( x )! Ad x c Px x ( )! x! ( x )! Now a 8 bc ( x + )! 8. ( x )! ( x + )! 8 x! ( x + )( x + ) 8 x. 5. (a) cocept.. (d) Without ay restrictio the 0 persos ca be raked amog themselves i 0! ways; but the umber of ways i which A is above A 0 ad the umber of ways i which A 0 is above A make up 0!. Also the umber of ways i which A is above A 0 is exactly same as the umber of ways i which A 0 is above A.. Therefore the required umber of ways (0!) 7. (c) The umbers betwee 999 ad 0000 are of four digit umbers. The four digit umbers formed by digits 0,,,,7,8 are 0 www.sakshieducatio.com P. But here those umbers are also ivolved which begi from 0. So we take those umbers as three digit umbers.

www.sakshieducatio.com Takig iitial digit 0, the umber of ways to fill remaiig places from five digits 5,,,7,8 are P 0 So the required umbers 0-0 00. 8. (c) The uits place ca be filled i ways as ay oe of 0,, or ca be placed there. The remaiig three places ca be filled i with remaiig digits i P 0 way. So, total umber of ways 0 80. But, this icludes those umbers i which 0 is fixed i 5 extreme left place. Numbers of such umbers P 5 0 0 Fix 5 P ways ways (oly, or ) Required umber of ways 80 0 0. 9. (c) Out of 7 places, places are odd ad eve. Therefore vowels ca be arraged i eve places i P ways ad remaiig cosoats ca be arraged i odd places i P ways. Hece required o. of ways P P. 0. (c) Words startig with A, C, H, I, N are each equals to 5! Total words 5 5! 00 The first word startig with S is SACHIN. SACHIN appears i dictioary at serial umber 0.. (a) Required umber of arragemets (Total umber of arragemets) (Number of arragemets i which N s are together)! 5! 0 0 0.!!!. (c) There are five seats i a bus are vacat. A ma ca sit o ay oe of 5 seats i 5 ways. After the ma is seated, his wife ca be seated i ay of remaiig seats i ways. Hece total umber of ways of seatig them 5 0. (d) 0,,,, 5, 7 : Six digits. The last place ca be filled i by,, 5, 7. i.e., ways as the umber is to be odd. We have to fill i the remaiig places of the digit umber i.e. I, II, III place. Sice repetitio is allowed each place ca be filled i ways. Hece the place ca be filled i ways. www.sakshieducatio.com

www.sakshieducatio.com But i case of 0 80 ways. Hece by fudametal theorem, the total umber will be 80 70.. (c) Give set of umbers is {,,...} i which 5 are eve six are odd, which demads that i the give product it is ot possible to arrage to subtract oly eve umber from odd umbers. There must be at least oe factor ivolvig subtractio of a odd umber form aother odd umber. So at least oe of the factors is eve. Hece product is always eve. 5. (a) No. of ways i which me ca be arraged at a roud table ( )! Now wome ca be arraged i! ways. Total Number of ways! 5!. (c) Required umber of ways 9!. W M M W W M M W W M W M 7. (a) The umber of ways i which 5 beads of differet colours ca be arraged i a circle to form a ecklace are ( 5 )!!. But the clockwise ad aticlockwise arragemet are ot. Hece the total umber of ways of arragig the beads (!) 8. (b) Fix up ma ad the remaiig me ca be seated i! ways. Now o two wome are to sit together ad as such the 7 wome are to be arraged i seve empty seats betwee two cosecutive me ad umber of arragemet will be 7!. Hece by fudametal theorem the total umber of ways 7!!. 9. (d) A garlad ca be made from 0 flowers i (9!) ways. 0. (b) Sice total umber of ways i which boys ca occupy ay place is ( 5 )!! ad the 5 girls ca be sit accordigly i 5! ways. Hece required umber of ways are! 5!.. (a) 8 differet beads ca be arraged i circular form i (8 )! 7! ways. Sice there is o distictio betweethe clockwise ad aticlockwise arragemet. So the required umber of 7! arragemets 50.. (d) It is obvious by fudametal property of circular permutatios. www.sakshieducatio.com

www.sakshieducatio.com. (a) Fix up a male ad the remaiig male ca be seated i! ways. Now o two female are to sit together ad as such the female are to be arraged i five empty seats betwee two cosecutive male ad umber of arragemet will be 5 P. Hece by fudametal theorem the total umber of ways is! 5 P 0 80 ways.. (a) It is a fudametal property 5. (c) Required umber of ways 7 7. {Sice the case that o fried be ivited i.e., 7 C 0 is excluded}.. (c) Here Cr Cr 8 ad C C r r+ 8. 0r ad 0r O solvig, we get 9, r. 7. (b) C ( ). 8. (a) It is obvious. 9. (b) ()! ( )!.!! ( )!! ()( )( ) ( ) ( ) Now 5 C r C C r or 5 5 5 5 50. (a) C r Cr+ C5 r Cr+ 5 r r + r. C r,. 8 8 8 8 8 5. (c) Required umber of ways C + C + C + C + C5 8 + 8 + 5 + 70 + 5 8 5. (c) We have places for teeth. For each place we have two choices either there is a tooth or there is o tooth. Therefore the umber of ways to fill up these places is. As there is o perso without a tooth, the maximum populatio is. 5 5. (d) The letters ca be select i C C ways. 5 Therefore the umber of arragemets are ( C ) 5! 5. (b) Total umber of hadshakes 5 C. ( ) 55. (b) C 5 5 8. C. www.sakshieducatio.com

www.sakshieducatio.com 5. (b) r r + 8 7 ad r r + 8 7 r r ( r + ) Or 9r + 9 r or r. So, 9 57. (c) Let there are cadidates the. C + C +... + C 5 5 8 8. + 58. (c) C. C ( + )!!.!.( )!!.( )! + 5. +.! 5 5 5 59. (d) A voter ca vote i C + C + C 5 ways.! 0. (c) The arragemet ca be make as. +. +. +. +. +. +. i. e., the ( ) sigs ca be put i 7 vacat (poited) place. 7 Hece required umber of ways 5. (a) Required umber of ways 5 9 C C C 5! 9!!! 5!. 9!!!!!!! (!) C C.. (c) The required umber of ways are ( 0 + )(9 + )(7 + ) 0 8 879.. (c) The required umber of words is ( C C + C C )! 9.. (b) Required umber of ways C + C + C + C + C + C. 5 5. (b) The total umber of two factor products 00 C. The umber of umbers from to 00 which are ot multiples of 5 is 0. Therefore total umber of two factor products which are ot multiple of 5 is 0 C. Hece the required umber of factors 00 0 C C 780. www.sakshieducatio.com

www.sakshieducatio.com p. (c) Sice umber of selectios are. Therefore the arragemet of r thigs ca be doe i Cr p p r!ways. Hece the total permutatios are C r! 7. (b) Required umber of ways 5 9 C C C7 0 70. 50 50 5 5 55 8. (b) C ( C + C + C +... C ) r p +. Takig first two terms together ad addig them ad followig the same patter, we get 9. (b) As for give questio two cases are possible. 70. (b) Expressio C r+ + Cr + Cr + Cr C r + + + + + Cr + Cr Cr+ Cr + + C r +. 5 + C, [ As C + C C ]. r r r 7. (c) The umber of times he will go to the garde is same as the umber of selectig childre from 8. 8 Therefore the required umber 5 C. 7. (d) We have ( )! ( k )!, 0 r ( r )! r! ( r )!( r + )! r + r + k +, k,, + k, + +, ;. + 5 7 7. (b) The required umber C C 7. (a) Sice the perso is allowed to select at most cois out of ( + ) cois, therefore i order to select oe, two, three,., cois. Thus, if T is the total umber of ways of selectig oe coi, the + + + T C + C +... + C 55..(i) Agai the sum of biomial coefficiets + C 0 + + C + + C +... + + C + + C+ + ( + + + C 0 + C + C +... + C ) + C... + C ( ) + + + + + + + + + + C+ + + + + ( T) + + T 8 + 55. www.sakshieducatio.com

www.sakshieducatio.com 75. (d) Word MATHEMATICS has M, T, A, H, E, I, C, S. Therefore letters ca be chose i the followig ways. Case I: alike of oe kid ad alike of secod kid.e. Case II: alike of oe kid ad differet i.e., 7 7! C C No. of words C C 75! Case III :All are differet i.e., 8 C No. of words 8! 80 0 8 7. (b) Required umber C 5 C C.. i, C No. of words C 8 77. (b) Sice the studet is allowed to select at most books out of ( + ) books, therefore i order to select oe book he has the choice to select oe, two, three,..., books.!!! Thus, if T is the total umber of ways of selectig oe book the + + + T C + C +... + C..(i) Agai the sum of biomial coefficiets + C0 + + C + + C +... + + C + + C+ + C... + C ( ) + + + + + + + + Or + C 0 + ( + C + + C +.. + + C ) + + C + + T + T + + ( ) + + 78. (b) Clearly, C T. + So, C C. + C + C ) C C ( or ( ) 7. 7. m+ + 79. (b) Total umber of poits are m + + k, the ' s formed by these poits k C Joiig poits o the same lie gives o triagle, such ' s are m C + C k + C m + k m k Required umber C C C C 80. (b) Sice C. +. 7 8. (a) Required umber of ways C C 0 5 85. www.sakshieducatio.com

www.sakshieducatio.com 8. (a) Case I: Whe A is excluded. Number of triagles selectio of poits from AB ad oe poit from AC + selectio of oe poit from AB ad two poits from AC m m C C + C C ( m + ) m..(i) Case II: Whe A is icluded. The triagles with oe vertex at A selectio of oe poit from AB ad oe poit from AC m. Number of triagles m + m ( m + ) m ( m + ) Required ratio..(ii) ( m + ). ( m + ) 8. (d) The required umber of poits 8 8 ( C ) C + C + C 8 + + 0 8. (c) Sice o two lies are parallel ad o three are cocurret, therefore straight lies itersect at C N (say) poits. Sice two poits are required to determie a straight lie, N therefore the total umber of lies obtaied by joiig N poits C. But i this each old lie has bee couted C times, sice o each old lie there will be poits of itersectio made by the remaiig ( ) lies. Hece the required umber of fresh lies is N C. C N( N ) ( )( ) C ( C ) ( )( ) ( )( )( ). 8 85. (c) Each set is havig m + parallel lies ad each parallelogram is formed by choosig two straight lies from the first set ad two straight lies from the secod set. Two straight lies m + from the first set ca be chose i C ways ad two straight lies from the secod set ca m + be chose i C ways. m + m+ m + Hece the total umber of parallelograms formed C. C ( C ). www.sakshieducatio.com

www.sakshieducatio.com 8. (c,b) 8 poits, 5 colliear : 8 5 (i) Number of lies C C + 5 0 + ' 8 5 (ii) Number of C C 8 0 80 s. 87. (b) Required umber of ways C 8 0 88. (a) Number of triagles C 0 89. (d) Required umber C C + 0 9 + 90 + 85. 90. (b) No. of triagles C. C. C 0 0 00. 9. (d) Alog horizotal side oe uit ca be take i (m ) ways ad uit side ca be take i m ways. The umber of ways of selectig a side horizotally is ( m + m + m 5 +... + + ) Similarly the umber of ways alog vertical side is ( + +... + 5 + + ) m. 9. (b) Total umber of rectagles Pr 9. (d) r! r C 9. (b) r C 5 7. me ad wome equal to 5. A group of 5 members make 5! permutatio with each other. The umber of ways to sit 5 members 5! Places are filled by 5 members by C 5 ways The total umber of ways to sit 5 members o seats of a bus 5! www.sakshieducatio.com C. 95. (b) Sice at ay place, ay of the digits, 5 ad 7 ca be used, total umber of such positive - digit umbers are. Sice we have to form 900 distict umbers, hece 900 7 5. 8 8 8 8 9. (c) The umber of selectios coefficiet of x i ( + x + x +... + x )( + x + x +... + x ).( + x)

www.sakshieducatio.com 97. (d) The umber of sub-sets of the set which cotai at most elemets is + + C0 + + C +... + C S (Say) + + + The S ( C + C +... + C ) 0 + + + + + + ( C 0 + C+ ) + ( C + C) +...... + ( C + C+ ) + + + + C 0 + C +... + C+ S. 98. (a) The total umber of digits are 9999 999 9000. The umbers of digits umber divisible by 5 are 90 0 800. Hece required umber of ways are 9000 800 700. 99. (b) The umber of triplets of positive itegers which are solutios of + y + z 00 00 Coefficiet of x i ( x + x + x +...) x. 7 00. (c) Sice 900 5 Hece, umber of divisors ( 7 + )( + )( + ) 8. www.sakshieducatio.com