05 - PERMUTATIONS AND COMBINATIONS Page 1 ( Answers at the end of all questions )
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1 05 - PERMUTATIONS AND COMBINATIONS Page 1 ( Aswers at the ed of all questios ) ( 1 ) If the letters of the word SACHIN are arraged i all possible ways ad these words are writte out as i dictioary, the the word SACHIN appears at serial umber 601 ( b ) 600 ( c ) 603 ( d ) 60 [ AIEEE 005 ] ( ) The value of 6 50 C 4 + r = r C3 55 C 4 ( b ) 55 C 3 ( c ) 56 C 3 ( d ) 56 C 4 [ AIEEE 005 ] ( 3 ) How may ways are here to arrage the letters i the word GARDEN with the vowels i alphabetical order? 10 ( b ) 40 ( c ) 360 (d ) 480 [ AIEEE 004 ] ( 4 ) The umber of ways of dtributig 8 idetical balls i 3 dtict boxes so that oe of the boxes empty 5 ( b ) 1 ( c ) 3 8 ( d ) 8 C 3 [ AIEEE 004 ] ( 5 ) A studet to aswer 10 out of 13 questios i a examiatio such that he must choose at least 4 from the first five questios. The umber of choices available to him 140 ( b ) 196 ( c ) 80 ( d ) 346 [ AIEEE 003 ] ( 6 ) The umber of ways i which 6 me ad 5 wome ca die at a roud table, if o two wome are to sit together, give by 30 ( b ) 5! 5! ( c ) 5! 4! ( d ) 7! 5! [ AIEEE 003 ] ( 7 ) If C r deotes the umber of combiatios of thigs take r at a time, the the value of expressio C r C r C r + + ( b ) C r ( c ) + 1 ( d ) C r + 1 [ AIEEE 003 ]
2 05 - PERMUTATIONS AND COMBINATIONS Page ( Aswers at the ed of all questios ) ( 8 ) If repetitio of the digits allowed, the the umber of eve atural umbers havig three digits 50 ( b ) 350 ( c ) 450 ( d ) 550 [ AIEEE 00 ] ( 9 ) If + 1 C3 = C, the the value of 3 ( b ) 4 ( c ) 5 ( d ) 6 [ AIEEE 00 ] ( 10 ) If C r - 1 = 36, C r = 84 ad C r + 1 = 16, the ad r are respectively 9, 6 ( b ) 9, 3 ( c ) 6, 3 ( d ) 6, [ AIEEE 00 ] ( 11 ) If ( 1 + x ) = C 0 + C1 x + Cx Cx, the the value of C1 C 3C3 C C0 C1 C C - 1 ( b ) ( + 1 ) ( c ) ( + 1) 1 ( d ) ( + 1) [ AIEEE 00 ] ( 1 ) A rectagle costructed of legths ( m - 1 ) ad ( - 1 ) uits where m, I ad small rectagles are cribed i it by drawig parallel lies. Fid the maximum umber of rectagles that ca be cribed i it havig odd uit legth. m - ( b ) m ( m + 1 ) ( + 1 ) ( c ) 4 m + - ( d ) m [ IIT 005 ] ( 13 ) If - 1 = ( k - 3 ) C r + 1, the k lies betwee ( -, - ) ( b ) (, ) ( c ) [ - 3, 3 ] ( d ) ] 3, ] [ IIT 004 ] ( 14 ) The umber of arragemets of the letters of the word BANANA i which the two N s do ot appear adjacetly 40 ( b ) 60 ( c ) 80 ( d ) 100 [ IIT 00 ]
3 05 - PERMUTATIONS AND COMBINATIONS Page 3 ( Aswers at the ed of all questios ) ( 15 ) Let T deote the umber of triagles which ca be formed usig the vertices of a regular polygo o sides. If T T = 1, the equals 5 ( b ) 7 ( c ) 6 ( d ) 4 [ IIT 001 ] ( 16 ) For r, r + r r - = + 1 r ( b ) r ( c ) r + ( d ) r [ IIT 000 ] ( 17 ) How may differet ie digit umbers ca be formed from the umber by rearragig its digits so that the odd digits occupy eve positios. 16 ( b ) 36 ( c ) 60 ( d ) 180 ( 18 ) If a = = r 0 1, the r = 0 r equals ( - 1 ) a ( b ) a ( c ) 1 a ( d ) oe of these [ IIT 1998 ] ( 19 ) A - digit umber a positive umber with exactly digits. Nie hudred dtict - digit umbers are to be formed usig oly the three digits, 5 ad 7. The smallest value of for which th possible 6 ( b ) 7 ( c ) 8 ( d ) 9 [ IIT 1998 ] ( 0 ) Number of divors of the form 4 + ( 0 ) of the iteger 40 4 ( b ) 8 ( c ) 10 ( d ) 3 [ IIT 1998 ] ( 1 ) A five digit umber divible by 3 to be formed usig the umerals 0, 1,, 3, 4 ad 5 without repetitio. The total umber of ways i which th ca be doe 16 ( b ) 600 ( c ) 40 ( d ) 315 [ IIT 1989 ]
4 05 - PERMUTATIONS AND COMBINATIONS Page 4 ( Aswers at the ed of all questios ) ( ) If C r stads for C r, the the sum of the series!! [ C C 3C ( - 1) ( +! where a eve positive iteger, equal to 1) C ], 0 ( b ) ( - 1 ) / ( + 1 ) ( c ) ( - 1 ) ( + ) ( d ) ( - 1 ) ( e ) oe of these [ IIT 1986 ] ( 3 ) Eight chairs are umbered 1 to 8. Two wome ad three me wh to occupy oe chair each. First the wome choose the chairs from amogst the chairs marked 1 to 4, ad the the me select the chairs from amogst the remaiig. The umber of possible arragemets 6 C3 4 C ( b ) 4 P 4 P 3 ( c ) 4 C 4 P 3 ( d ) oe of these [ IIT 198 ] ( 4 ) Te differet letters of a alphabet are give. Words with five letters are formed from these give letters. The, the umber of words which have at least oe letter repeated 69,760 ( b ) 30,40 ( c ) 99,748 ( d ) oe of these [ IIT 1980 ] ( 5 ) The value of the expressio 5 47 C 4 + j = j C3 equal to 47 C 5 ( b ) 5 C 5 ( c ) 5 C 4 ( d ) oe of these [ IIT 1980 ] ( 6 ) C 36 C 84 ad r - 1 = r = + 1 = 16, the r 1 ( b ) ( c ) 3 ( d ) oe of these [ IIT 1979 ] ( 7 ) There are 7 poits i a plae. 5, 10 ad 15 poits are colliear o dtict lies. By joiig these poits, how may dtict lies ca be formed? 194 ( b ) 170 ( c ) 435 ( d ) oe of these ( 8 ) I the above Q. 7, how may dtict triagles ca be formed whose vertices are the give 7 poits. 7 C 3 ( b ) 300 ( c ) 30 ( d ) 340
5 05 - PERMUTATIONS AND COMBINATIONS Page 5 ( Aswers at the ed of all questios ) ( 9 ) The umber of ways of puttig 10 differet thigs i boxes such that there are ot less tha thigs i ay of the two boxes 104 ( b ) 103 ( c ) 1013 ( d ) 100 ( 30 ) The product of r cosecutive positive itegers divided by r! a proper fractio ( b ) a positive iteger ( c ) r ( d ) oe of these ( 31 ) If C r - 1 = 36, C r = 84 ad C r + 1 = 36, the r = 1 ( b ) ( c ) 3 ( d ) 4 ( 3 ) 30 C C C C C 13 = 0 ( b ) 3 C 13 ( c ) 33 C 14 ( d ) 3 C 14 ( 33 ) A polygo has 54 diagoals. The total umber of dtict triagles that ca be formed usig its vertices 0 ( b ) 165 ( c ) 86 ( d ) 16 ( 34 ) A set of 5 parallel lies with dtaces 1,, 3, 4 betwee cosecutive lies itersects aother set of 5 parallel lies oblique to the first set with dtaces 1.5,.5, 3.5, 4.5 betwee cosecutive lies. The umber of rhombuses formed equal to 1 ( b ) ( c ) 3 ( d ) 4 ( 35 ) Four dice are rolled. The umber of possible outcomes i which at least two dice show 6 16 ( b ) 900 ( c ) 150 ( d ) 171 ( 36 ) Six poits i a plae are joied i all possible ways by idefiite straight lies. No two of them are coicidet or parallel ad o three pass through the same poit ( with the exceptio of the origial six poits ). The umber of dtict poits of itersectio equal to 105 ( b ) 45 ( c ) 51 ( d ) oe of these
6 05 - PERMUTATIONS AND COMBINATIONS Page 6 ( Aswers at the ed of all questios ) ( 37 ) 6 me ad 4 wome are to be seated i a row so that o two wome sit together. The umber of ways they ca be seated ( b ) 1780 ( c ) ( d ) ( 38 ) A test costs of 10 multiple choice questios each havig four alterative aswers of which exactly two are correct. A studet has to mark two aswers ad h aswer cosidered correct oly if both the selected aswers are correct. The umber of ways of gettig exactly 8 correct aswers by a studet aswerig all the questios 115 ( b ) 405 ( c ) 180 ( d ) oe of these ( 39 ) 10 boys ad 10 girls sit alterately i a row ad the alterately alog a circle. The ratio of umber of ways of sittig i a row to the umber of ways of sittig alog a circle 5 ( b ) 10 ( c ) 15 ( d ) 0 ( 40 ) The sum of all 4 digits that ca be formed by usig the digits, 4, 6, 8 allowig repetitio of digits p ad without allowig repetitio of digits q. The ratio of p to q 3 3 ( b ) 16 3 ( c ) 64 3 ( d ) 16 Aswers a d c b b b b c c b d d d a b c c c b a c e d c c c a d d b c a a c d c a a d a
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