4. PERMUTATIONS AND COMBINATIONS

Transcription

1 4. PERMUTATIONS AND COMBINATIONS PREVIOUS EAMCET BITS 1. The umbe of ways i which 13 gold cois ca be distibuted amog thee pesos such that each oe gets at least two gold cois is [EAMCET-000] 1) 3 ) 4 3) 1 4) As : 1 Sol : Requied umbe of ways coefficiet. x 13 i (x +x 3 +.) 3 coefficiet. x 7 i (1+x+x +.) 3 coefficiet. x 7 i (1+x) -3 9 C 7 9 C 3. If C (, 3) : C (, ) 1 : 1, the [EAMCET-000] 1) 4 ) 3) 4) 8 As: Sol : C 3 : C 1 :1 C 3 1. C c ( 1)( ) ( ) The umbe of quadatic expessios with the coefficiets daw fom the set{0, 1,, 3} is 1) 7 ) 3 3) 48 4) 4 [EAMCET-000] As : 3 Sol : ax +bx+c 0 a ca be filled i 3 ways b ca be filled i 4 ways c ca be filled i 4 ways Requied o. of ways 3 x 4 x The umbe of ways i which boys ae 4 gils sit aoud a cicula table so that o two gils sit togethe is [EAMCET-001] As : 1 1)! 4! )! 3! 3)! 4) 4! Sol : Fist we aage boys aoud a cicle i (-1)! 4! Ways the we have gaps betwee them the aage 4 gils i gaps aagemet of 4 gils i gaps Aagemet of 4 gils i gaps P 4! Requied o. of ways! 4! 1

2 Pemutatio ad combiatio. Usig the digits 0,, 4,, 8 ot moe tha oce i ay umbe, the umbe of digited umbes that ca be fomed is [EAMCET-001] 1) 1 ) 4 3) 10 4) 9 As: 4 Sol : Requied o. of ways! 4! If ad ae iteges such that 1 < < the. C (-1, -1) [EAMCET-00] 1) C (, ) ). C (, ) 3) C (, ) 4) ( - 1). C (, ) As: 3 Sol :. c( 1, 1).( 1) c 1 ( 1 )! ( 1 )(! )!.!.!( c. c(,) )! 7. The least value of the atual umbe '' satisfyig c(,) + c(,) > c(+1,) [EAMCET 00] 1) 10 ) 1 3) 13 4) 11 Sol : Give c + c ( + 1) > ( + 1) ( + 1) c > ( + 1 )! ( )! >10 c c ( + ) 1! >!! ( 4)! The least value of is The o. of ways such that 8 beads of diffeet colou be stug i a eckles is...[eamcet-00] 1) 0 ) 880 3) 430 4) 040 Sol : Requied umbe of ways ( 8 1 )! 0 9. The umbe of digited umbes which ae ot divisible by ad which cotais of odd digits is [EAMCET-00] 1) 9 ) 10 3) 4 4) 3 Sol : The odd digits be 1,3,,7,9 Requied! 4!

3 3 Pemutatio ad combiatio 10. Let l 1 ad l be two lies itesectig at P. if A 1, B 1, C 1 ae poits o l 1, ad A, B, C, D, E ae poits o l ad if oe of those coicides with P, the the umbe of tiagles fomed by these eight poits. [EAMCET-003] 1) ) 3) 4 4) 4 As: 4 Sol : If tiagle is icludig poit P the othe poits must be oe fom l 1 ad othe poit fom l, Numbe of tiagles fomed with P. 3 ( E ) c c Whe p is ot icluded Numbe of tiagles fomed 3 3 ( ) + E c c c c Total umbe of tiagles (E 1 ) +(E ) The umbe of positive odd divisos of 1 is [EAMCET-004] 1) 4 ) 3) 8 4) 1 Sol: The factos of The odd divisos ae the multiplied 3. The umbe of positive odd divisos A thee digit umbe is such that the last two digits of it ae equal ad diffeet fom the fist. The umbe of such s is [EAMCET 00] 1) 4 ) 7 3)81 4)900 As: 3 Sol: If the last two digits ae equal to the the fist digit may 1 to 9 If the last two digits ae equal to 1 to 9 the the fist digit may be selected i 8 ways. The equied umbe If N deotes Set of all positive iteges ad if ad if is defied by the sum of positive divisos of. the whee is a positive itege is [EAMCET-00] 1) k ) ( k ) 3) ( k ) 4) ( k ) s: 3 Sol: Give f(x) the sum of positive divisos of

4 k 3 k (.3) 3( ) f k + 1 1( 1 3 ) 1 k ( ) Pemutatio ad combiatio 14. The umbe of atual umbes less tha 1000, i which o two digits ae epeated is [EAMCET 00] 1) 738 ) 79 3) 837 4) 70 Sol : The umbe of 1 digit umbes 9 The umbe of digit umbes 9 x 9 81 The umbe of 3 digit umbes The umbe of Requied umbes The umbe of ways of aagig 8 me ad 4 wome aoud a cicula table such that o two wome ca sit togethe, is [EAMCET-007] As: 1) 8! ) 4! 3) 8! 4! 8 4) 7!. P As: 4 Sol: Numbe of ways of aagig 8 me aoud a cicle (8-1)! 7! The we have 8 gaps betwee them Numbe of ways of aagig 4 wome i 8 gaps 8 p 4 Requied umbe of ways 7!. 8 p 4 1. If a polygo of sides has 7 diagoals, the [EAMCET-007] 1) ) 3 3) 0 4) 1 Sol: Numbe of diagoals of a polygo of sides 7 ( 3 ) 7 ( 3 ) 0 ( 3 ) balls ae to be placed i 9 boxes, ad of the balls ca ot fill ito 3 small boxes. The umbes of ways of aagig oe ball i each of the boxes is [EAMCET-008] 4

5 1) 1870 ) ) ) 1780 As: 3 Sol : balls ca be placed i boxes (othe tha the 3 small boxes) i p ways The emaiig 4 balls ca be placed i the emaiig 4 boxes i 4! ways. The equied umbe of aagemets p 4! 18. If p 3040 ad c the the odeed pai (,) Sol : 1) (1,) ) (10,) 3) (9,4) 4) (1,7) As: p c 3040! 10!! p 3040 (,) (10,) 10 p 10 Pemutatio ad combiatio 19. The umbe of subsets of {1,,3, 9} cotaiig at least oe odd umbe is [EAMCET-009] 1) 34 ) 39 3) 49 4) 1 As: 3 Sol : No of subsets P poits ae chose each of the thee coplaa lies. The maximum umbe of tiagles fomed with vetices at these poits is [EAMCET-009] 1) p3+3p 1 ) ( p 3 + p) As: 4 Sol : Let the lies be L 1, L, L 3 3 p p p Max o of tiagles ( ) 3 c c c1 + c p p + p p 3p 3 + p ( p 1) 3 ( ) 1 3) ( p 3) p 4) p ( 4 p 3) p (4p-3) 1. A biay sequece is a aay of 0 s ad 1 s the umbe of -digit biay sequeces which cotai eve umbe of 0 s is [EAMCET-009]

6 Pemutatio ad combiatio 1) -1 ) -1 3) ) As : 1 Sol : If is eve, o of -digit biay sequeces -1.