4. PERMUTATIONS AND COMBINATIONS Quick Review

Transcription

1 4 ERMUTATIONS AND COMBINATIONS Quick Review A aagemet that ca be fomed by takig some o all of a fiite set of thigs (o objects) is called a emutatio A emutatio is said to be a liea emutatio if the objects ae aaged i a lie A liea emutatio is simly called as a emutatio 3 A emutatio is said to be a cicula emutatio if the objects ae aaged i the fom of a cicle (a closed cuve) 4 The umbe of (liea) emutatios that ca be fomed by takig thigs at a time fom a set of dissimila thigs ( ) is deoted by o (, ) o 5 The umbe of emutatios of dissimila thigs take at a time is equal to the umbe of ways of fillig of blak laces aaged i a ow by dissimila thigs 6 Fudametal coutig icile : If a oeatio ca be efomed i m ways ad a secod oeatio ca be efomed i ways coesodig to each efomace of the fist oeatio, the the two oeatios i successio ca be efomed i m ways 7 If i k oeatios, the fist oeatio ca be efomed i ways, the secod oeatio ca be efomed i ways, thid oeatio ca be efomed i 3 ways ad so o, the the k oeatios i successio ca be efomed i 3 k ways 8 = ( )( ) ( + ) 9 If is a o-egative itege, the factoial is deoted by! o ad defied as follows (i) 0! = ; (ii) If > 0 the! = ( )! 0 If is a ositive itege, the! is the oduct of fist ositive iteges ie,! = 3 =! ( )! The umbe of emutatios of dissimila thigs take all at a time is =! 3 ( ) ( ) = + 4 The umbe of ijectios (oe-oe fuctios) that ca be defied fom a set cotaiig elemets ito a set cotaiig elemets is 5 The umbe of bijectios (oe-oe oto fuctios) that ca be defied fom a set cotaiig elemets oto a set cotaiig elemets is! 6 The umbe of emutatios of dissimila thigs take at a time whe eetitio of thigs is allowed ay umbe of times is 7 The umbe of emutatios of dissimila thigs take ot moe tha at a time, whe each ( ) thig may occu ay umbe of times is 8 The umbe of fuctios that ca defied fom a set cotaiig elemets ito a set cotaiig elemets is

2 emutatio ad combiatio 9 The umbe of emutatios of thigs take all at a time whe of them ae all alike ad the! est all diffeet is! 0 If thigs ae alike of oe kid, thigs ae alike of secod kid, 3 thigs ae alike of thid kid ad so o, k thigs ae alike of k th kid i k thigs, the the umbe of ( k )! emutatios obtaied by takig all the thigs is!!! The umbe of cicula emutatios o diffeet thigs take at a time is The umbe of cicula emutatios of diffeet thigs take all at a time is ( )! 3 The umbe of cicula emutatios of thigs take at a time i oe diectio is 4 The umbe of cicula emutatios of thigs take all at a time i oe diectio is ( )! (+ ) 5 (i) + = (ii) (iii) = = + 6 If is a ositive itege ad is a ime umbe the the exoet of i! is whee [x] deotes the geatest itege x 7 The umbe of ways i which m (fist tye of diffeet) thigs ad (secod tye of diffeet) thigs (m + ) ca be aaged i a ow so that o two thigs of secod kid come togethe is (m+ ) m! 8 The umbe of ways i which m (fist tye of diffeet) thigs ad (secod tye of diffeet) thigs ca be aaged i a ow so that all the secod tye of thigs come togethe is! (m+)! 9 The umbe of ways i which (fist tye of diffeet) thigs ad (secod tye of diffeet) thigs ca be aaged i a ow alteatively is!! 30 Sum of the umbes fomed by takig all the give digits (excludig 0) is (Sum of all the digits) ( )! ( times) 3 Sum of the umbes fomed by takig all the give digits (icludig 0) is (Sum of all the digits) [( )! ( times) ( )! ( ( ) times)] 3 Sum of all -digit umbes fomed by takig the give digits (without zeo) is (sum of all the digits) ( times) 33 Sum of all the -digit umbes fomed by takig the give digits (icludig 0) is (sum of all the digits) [ ( times) { ( ) times}] k

3 emutatio ad combiatio 34 The umbe of ways i which m (fist tye of diffeet) thigs ad (secod tye of diffeet) thigs, (m ) ca be aaged i a cicle so that o two thigs of secod kid come togethe is (m )! m 35 The umbe of ways i which m (fist tye of diffeet) thigs ad (secod tye of diffeet) thigs ca be aaged i a cicle so that all the secod tye of thigs come togethe is m!! 36 The umbe of ways i which m (fist tye of diffeet) thigs ad (secod tye of diffeet) thigs ca be aaged i the fom of galad so that all the secod tye of thigs come togethe is m!!/ 37 A selectio that ca be fomed by takig some o all of a fiite set of thigs (o objects) is called a combiatio 38 Fomatio of a combiatio by takig elemets fom a fiite set A meas ickig u a elemet subset of A 39 The umbe of combiatios of dissimila thigs take at a time is equal to the umbe of elemet subsets of a set cotaiig elemets 40 The umbe of combiatios of dissimila thigs take at a time is deoted by C o C(, ) o C o 4 C =!!( )! 4 ( )( )( ) C = + =! 3 43 C = C 44 ( ) + C 45 If C = C s, the = s o + s = ( 3) 46 The umbe of diagoals i a egula olygo of sides is C = 47 The umbe of ways i which (m + ) thigs ca be divided ito two diffeet gous of m ad ( m + )! thigs esectively is m!! 48 The umbe of ways i which thigs ca be divided ito two equal gous of thigs each is ()!!(!) 49 The umbe of ways i which ( + + k ) thigs ca be divided ito k diffeet gous of ( k )! thigs, thigs, 3 thigs, k thigs esectively is!!k! 50 The umbe of ways i which k thigs ca be divided ito k equal gous of thigs each is (k)! k k!(!) 5 The total umbe of combiatios of ( + q) thigs take ay umbe at a time whe thigs ae alike of oe kid ad q thigs ae alike of a secod kid is ( + ) (q + ) 3

4 4 emutatio ad combiatio 5 The total umbe of combiatios of + q thigs take ay umbe at a time, icludes the case i which othig will be selected 53 The total umbe of combiatios of ( + q) thigs take oe o moe at a time whe thigs ae alike of oe kid ad q thigs ae alike of a secod kid is ( + )(q + ) 54 The total umbe of combiatios of ( k ) thigs take ay umbe at a time whe thigs ae alike of oe kid, thigs ae alike of a secod kid, k thigs ae alike of kth kid, is ( + ) ( + ) ( k + ) 55 The total umbe of combiatios of ( k ) thigs take oe o moe at a time whe thigs ae alike of oe kid, thigs ae alike of a secod kid, k thigs ae alike of kth kid, is ( + ) ( + ) ( k + ) 56 The total umbe of combiatios of diffeet thigs take ay umbe at a time is 57 The total umbe of combiatios of diffeet thigs take oe o moe at a time is 58 C + C + C + C = 0 α α αk 59 If is a ositive itege, the ca be uiquely exessed as = k whee,, k ae imes i iceasig ode ad α, α, α k ae o-egative iteges This eesetatio of is called ime factoisatio of i caoical fom o ime owe factoisatio of α α αk 60 The umbe of ositive divisos of a ositive itege = k (the ime factoisatio) is (α + )(α + )(α 3 + ) (α k + ) 6 (i) C = (+ ) C (ii) = + (iii) C = Cs = s o + s = + 6 (i) = (ii) = (iii) = C ( ) C ( ) 63 The umbe of aallelogams fomed whe a set of m aallel lies ae itesectig aothe set m of aallel lies is C C 64 If thee ae oits i a lae o thee of which ae o the same staight lie excetig oits which ae colliea, the (i) the umbe of staight lies fomed by joiig them is C + C (ii) the umbe of tiagles fomed by joiig them is C3 C3 65 The umbe of ways that soveeigs ca be give away whe thee ae k alicats ad ay (+ k ) alicat may have eithe 0,,, 3, 4, 5 o soveeigs is Ck 66 (i) The umbe of ways i which exactly lettes ca be laced i wogly addessed eveloes whe lettes ae uttig i addessed eveloes is ( )!! 3!! (ii) The umbe of ways i which diffeet lettes ca be ut i thei addessed eveloes so that all the lettes ae i the wog eveloes =! ( )!! 3!! 67 (i) The umbe of ways of asweig oe o moe of questios is (ii) The umbe of ways of asweig oe o moe of questios whe each questio have a alteative is 3

5 emutatio ad combiatio (iii) The umbe of ways of asweig all of questios whe each questio have a alteative is k k k 68 The umbe of distict ositive itegal divisos of whee,,, ae imes i ascedig ode, is (k + )(k + ) (k + ) k k k 69 The sum of distict ositive itegal divisos of whee,,, ae imes i ascedig ode, is k + k + k + 5