F CHAPTER=5 peatutyslom _- vi Se AND & yt COMBINATIONS | i Becrutotion’ Peemutation mvans both election ond csongerents lof things: GF Forming tuo digis numba from the guen digits 43/3 Fe 19,13, 81, 93, 2), 3a. | cornbicotion- Combiration meors only election of thing foctoriad Hotation Basicall Of=) LIfe) in gerecal n= nwo--} Be axirg = BXAKISG p= Unga By S}= Sry, = lao 6 = Groxux3xaxl =.490 Fucdomental painciple of Counting: 7fa Uosk teh an'be peofonrred pre! diffesent ways Cand a bécond UERK Wa con be peofoomed in n different LCOS, then the ttoo wasks con be pesfosmed in'rery JFfeoarst ways: | Gi fa man has threat hio6 Ta, 13 and tao) Fors Ti-To a thenshe tan celle’ eekecled pais3ax3 gies AFfecent Ways as, eypoired belo: mT RY, bh 1 Bh Ble Uneos Peomutotion Fema qe finite Set of elarenis selects Some on all of them and cworping them in a tine is called a ines peomutation [és Simply’ permutation») en 1 pemutation:- From ‘a. qven Finite set of thi choosing conne os all of then ond awe orgirg thro round a, Circle ig called 0 “coculaa peomutation’ &- Somme OF the Ceculor permutations formal usingtne digts h234V%e- SP LG.oN Storey. \ Parnctottens fn! pin oid) ilows. ries. thea 3 tokens ot 0 +4), IE, 6 owe ‘posithe inte 5 and BAN “ten of primustatiers of in‘dissimibss thi = sata (ita devoted by 08 "by itis defined 03 ("is = . Por-1 . P bility Conditions i iB hy mt ible_by au TF the digits in the lost place “‘ofagiven \ nurbe iS 0 Os even nurnbess they the | given ‘nunbes: ts _| divi ibe bya &r 520,62, 16+ j | Divisible ty 3: the Suro of the dys. in the pom ee | Ny amultipe Je of 3 aes given numbes is divisible | H ab Dvsibe we” + 381 J 34841 slDisamultiple of gpthe number tthe bat 400 places Sa maltifle of (| number is divisible by u+ i &» 520: i hiwDivicibsle je by Sr BE the digits in ae iste Omy te dist Be , iS ohuw i | te qu tn is a \ A | Yas, Sto: inthe {ost tao a ies fa «| >. pg, = taKx1o Q-@-2 co tacBtis ] ie TE nia ove positive integers: vond wn: . | I b Mp =1 ne | di Open] ' : i \ WP) =n ! a © ye) ty ps = nt : | v 0! | W a Pr ent i evens) | i oD 2m i w Pye CRS wh"? uieeae ‘then the qivey | i, we we ae by ss: noe the nage \ gen nuerben We erlliple FS a t £927 BAB MT on ne it 5KPrecise 5 i io “AECiSe BE), {1-01 A) cs # if "Py = 1320, find n. ica gt np, 2 |38R10 S } >, flies MX XO (Cap) "Pe Dw-DM2) Mp= ps Oe . OD! | TPP, =42-"B, Find n. = feofobdoprnfrn-s)tn-ee cong dtoArtnbYpu) N=D(12=2) 212x108!) 4 (-Dln-6)= yo OD0-6)= 16 n-5=7 n-6=6 5 HOM Pe = 8:9 finda. ay OU WADNAIIp3) 8 FoF) 0) GAD trans” 2 mo L 8 @-win-5) Tint)= 2(n—u)(n-5) Tne t= es En-un+20) an gn~Ber UO A971 =0 Ane asnt 32-0 ani —gan-3n3-0 2n(0-1)-30N-11)=0. (na (8n-D)=0 nell an=a we° ith (9-1 2 Mi cormprirg) vorlh Ua Sei Px =8, 5] TFB, 4, Qi Toa Feet) A Finds. Sol- V7 Bo ost L Fenn Peo)— meg Be. mal Wis sb aa eat SM be he odmit his cons» ihe Echedle SO jyatro “OF them wall be nthe come chao! J} plumbee of Schools trys 5 Aumbe oF Eons (5 "bY 2. Nurbes OF tonys OF Joining U cong in 5 ochess "Pe BR ue =120 at Pap thece ove 35 aavkoay glations on a vailesay line, hoo mony ype of Single Second: Clase lickels muck be poinied, co aso eravie ey to-teavel fom ene Slation to anathea: *, Aumbes of wailooy alelions (m= 35 ; furrloee of Gecerd Class dickels must be pointed fee .a Sing ‘pened 40 one Stotion to onctheo slalion v2) "Ps = Fp, = asl @S-2u) |} an ) bacrcot = Boyar apy! Sef Fra. class thee exe 30 Studers: Ohthe neo gen oy. excny etedlerts peels a ting cox to oll his/hes clossmertes: Find the froted nursbes of greeting doers pesied by hem. ; Hy Aumbes of etuden's (n)= 30 i [the toto! noo gueling mada gested bythe 20 students 1s ' : Mp ,= 200 “Bot. 30x20x76) Po= Pe @o-2)! 2B = 30199 Weel sine Pt = Bio. = Find the Mumbes of os ef emnenping the Teka: of the nod LrRiMlg(e so that the celatve positions of tha yowels end he leoreorants axe not disttbed, Ko. of vols a=3 (fie CALE 1) Ao-of conden (1/R, M4.) =S Tota No-of tettexS=gZ apes Ve, frst woe emsange tree \ouxls in, tee veative pores 'S °P; 3! Second ord then we GeDdrge five corsbnants in thes five velative moan bes Uemns on} thee one § schoo “U iI BOC - = Iplaces iS Spero 5! 1 5 The total no-of coonfrenis =3)x5) =H90: ' «pind the gana ‘ / s‘ Note :- | eum of ‘oll De ci nurbes that con be fom 089 the Hy “ny” non-2ew digits 1) ; Tor Powe pe Semof the digits xn (ste) othe Cathy oF the’ ead digis nimsbed’ Incliding “0! C200) tint Os be foomed a9 ean 'n’digits IS. Roo, Sum of the digits nt {tines} = + N@D tress O-D, x Sumof-the digits x lll Rea T Bad the Sum ef et y digited rurrbas that anbe fomed Sig the digite 02.478 caltiout agg tion Given digits 0,9,u,7,8 n=5 . v4 hot y the .saly oF all 4 | digs numba rot on be formed “using hy irdigls 15-9 v ; fom 8 x Sum\et te des R= ~ ede 1st} +o) j {oh Fo Of-the digits xi s-.-=--. ‘enn ties} \ Me,% arvntl praia) p ont Son eh = BBM =~ NSIE \ » Gath 5 26 . 5 Seghia! : = 5,45, 958. é J.aMe ve 7305 aS) TOOT aye ete s aA] Find the cum of at 4- dij murbs tt con be Moe ing the digs ,3,5,7.q. f ' . Dlr Given. digits 1, 3,S,1.9, h=5 i ey “he Gum oF all w digs numbes that con be fosmed using the digits is = Ml * Sum of the digi rin este see 1 (times) Poy = Up, x ASX Ui , = Suxdsaii) ; 4 ~ 600xti 1 3 = WEAR. 6, 66, 6a y 159 = £221) TF PR =1680, find n, Sl: mal 1680 Mp y= 168KI0 "ps Bry El adaere, n= Loo? = NM-NN-2)(N-3) = 8(8-D@- 2) (8-3) =OXTXE KS Hog n=g. Al ‘ga eee Te Fae Pe ! 2) Pp. 300 2p. Baxlo i "Ps= mt 2p, “Loe 25 IF be a) P = 208: con] og Ore foray 00 | Pests) _ 3 Weed © | => Bebe. p Pots) p> —Wet ns Go| San 56x55 KS! or ofa) eseen i iea Tit ce €93:-|g¢OHDp . i 12 | TOR p=: Finan. ) | pir} +Dp | S23 a "Ps : ani al= 9" inn KOA) _ l (WADE Dw-S)p-uy 2 Bint 1)= 3 (ry) Q42=3n-42 AP ay BN-3N-19-2=5 ‘ n-lu=0 pole aon” 4 “sp 3-| Fird the cum of all u digital numbes that the dig’® 43,U,5,6 ole weet Grete oe shy ep\") Glen digits Vause n=5 i wey | athe sum of all a dit numbess hot cary be feed using th digtis ing the |e ba Pop So ofall the digs Xi - -~ Veotinres) |= Sp, aban | e080 “ead” @ = gp xeBe iH 1B X N11 per = 43axiil =4TAIS2= $9" | Find the Dunes of numbex, that abe: ‘Speattes thors vemmpechice ton be foomed using the digitse.9,4,6,8: with out, oepetition. eplt| G00 digits 09,4, 68 ie : Now, Loe find the nunobeo of numbers cohich ove peas than Yodo, fsons the ‘folbusirg 400 cases. cases digs = Yoo : "6s < & e50.4s, Hee NEY p 1 Frost, we aocorge “the fiest ploce Bat 16/8 in “thse jord then the reat three places con be axeongee|votth: varaiining Rew dgts in 4 uonye- J sthe tolod oot u digits = 2% 4p, “YP |rst, \oe cmsarge the fast pace with 9L U/e/B in: Y OO,8 ong o.0. [tn the next fous places aon be aseanged with remaining Bi P| digits in ow : - Phe total nurnbewk 5 digi =UeUp, =Uxyl i 4, =ARY : = 96 | the total nurnbes which ace gpartes than Woop is 72+96= 168, | Find the numbes of war of cssonging 5 diffeent aaieican . backs, ty diffetent pid books ond’ 3 ciffecent chemisty \|bedks’ Such that the books ofthe sorre Subject ose get, — i+] Given, mathervotica beoks = 5 physics Books © = 4 chemistry books = 3 , 5 diffeset Mathematics books dan be coon; emoq ing 9. Foctosiod wotp. (5!) ater Ypres books, con be oseargedl orverg aMemet nu) 2a, 3 diffecent chemistey books con be amanged arora themsds in 81 wor: the Novel ecoys of coxongng Mathematics, prysits, chemist teakecty yaaa ne i c 4 jhe Rival’ amscrgprers=31* BRGY RB) SG xXtdOXdU XG > = 103680 . pind the nurebes of 5 lettes wode that can be foomed usi a the letters of the ond CONSIDER: How mony) of them begin with “cy hao of then, end .wolth "Rand ow many of them begin wsith*e ” ond end with 'R" SO} the usosd consiner hoe 9 tetlers, fn-8 %-S] the five lettes wootd in Bp ways Bp = BXIx6 x5 XY = 6120 wy Begn edith? Ct ! C v pest place “econ ke filed with ene Uy sercoining ploces, con be filled usth semaining = “leHess 6'Py LOY, athe No-of five (ete uxed begin with c = 1h y = Ux6x SKY 9 a0 ' 7 {ae BYoro} or & Eno wsth'R COTTE] test place “e" aan be filled 1 unk ne ing on | aces. con be, Fly saith zemaining, 5 letersi Ip, , woop: “the rlo-of five lees waded end with rep ei : 4 = V6 XSxy { =Bu9.ae Begin with “cond end wi Rh TRI et ps A with & Lace Rate: Fiadt pice “econ be filled wait cre. wan ond. look Pi filed est one coay "ROY be Ried toith vereatnng 6 leHesy in Bps, voays: [The ov oF 5 tetlessuensl begin withc"and end uth” R= OP, { 9 | gind the hurkes of coy, of cxnorging tthe Lette of the woed .| Moxy co that no vouR) ascuples even ploce- CELE] I Given weed. MONDAY No-of wowel = 8 No-of consonants = uy Since, no vewrels occupies even place the tex vourls con be Filled uatth 3 odd places in a waoys- Sroting “places, con be fnted estth yconsorants inp, ry 2 The Ao-of Reyuized oxeargements=3 p x uy =6xQy4 S14 a.| tind the ruts of Lnays of seating © Studenis 4p Ab,- -- Ay In anew euch thot, bialth 2 p { r A fA Aa; Azsit to at Wb A Aa, Ag sit ina Specfied codes. tid Ay Aa, Ag sit together inc. Specified coclen. co\h Concides Ars Aa,As) a8! ene unit and Ay, Ag-- Units. this 6 unilS’ con be ossorged jn 8! Lays: the students Ay, Ag, As can be awzanged orrorg cteréelf in 3) UND + s the rlovof Requined omonge ment = 81X31 AL, Ay, As, Sit INO. specified osteo.” So Fe “ George Ay, Aa, Aa in Specified osdeo Avy Ag, As com be exrorged in Dp, ‘St Remaining 4. peasms can be emoanged 1! oe ft when Ut Roquived cenorgerents= IOPS Bert /s: Zi 8 Aig O84Hl doge thes i “Speci Fi : diy AL An, Aa 's s in ox ified _oedeo! 4 Conese A Aa, Aa in that osdes hes one unit » Remoini ; ave units This 8 unils con be asanged in 8) oe 2 the Novf Requioed exsongennents = =8} unys: 2: [pind the minnbed oF ways in which 5 ved Eas, 4 Bock bay lof diffesent Si2€ can be canarged ind ToD SO tat % Motuy Y, bas: of he Sarre colo come aythe bolls of the Some colours Comme tos SO Jt) no tuo balls of the some _cobus ame togethea:> ty black balls can be “awsonged belwoeen 6, 26d balls in) u) tooyp. 5 ved balls con be assanged ariong themselves in 5) wa + The toto! No-of oeon = YRS] % I) the. balls of the some cobuy! corr core together :— = 5 Red balls con be toeat a9 “one we unit and gq Heck Bal an | ke foeat 03 one unit This “tS unis’ aan ‘be owuonged +80 3] t 5 Ral balls con be amonged orrong therseth in 5) wos Ut bock balls com be axwanged Ormeng “themself in Uf cons. + the fot. No: of Requined crcongere nts < uy. | pind he nia oF undigit numbers ‘that! dor le eared | uaing the digits 18,5,6,7,, if ro tae “digit ont th Séme! txo | of-tiem ove divisible ‘by: (29 di hyo divs yey Ass XS)xy] Spr Are moot w digits HS nunibes, that an be fred si ween fihot con be net apg te digits Beet) S69 is yz Sh e000 « ‘ dy_Divisible by"9")~ - ; Teal lost pace mast be fitel with 30s 6 iF ap cay Demanirg thee places Filled usith Soa a digits’ in Up. eaoys. othe No of 4+ dys nombess * divicible bya 2 axup, “sacl Sax2y » SUB:77) jy Dvibe betatie west arenes oof pal qrmbe iS dvb by 3 only ushen “the sum of digits in spat nuebas TS rruttile of 3 the Gog digs such “that thee Sum ig rattle oP 3 fears te Quen digitsase *,8,57 (15). js the MOoF u digHS Nunbees divicible bya apy =ul jv Dvicible by us : ae & = lost hoo place, must be fj (ss. led with 19.00 16 Go 58 63 S6.0071260 Ter: in6 oe Reweining & Placed con be filled wilh remoining 3 ag® 2p wots. 2 the, Roof uy digttMunbes divisible by u> 6x3, =6r 3) ete ee, = 6x6 pups by o- d.08 lost Ploce must be filled sth & in 1 spamoinirg 3 places Bo be Filled with semoining uy digs 19 Up, ang: ‘The: Nock ey digit’ thumbes divisible bys wb Miss> a losk ero places must be filled with 95 00 in 2ecayE, oematning ® places 49 con be filled twoith Deraining 3 digits is 3p, way, tthe Nov of y digits numben ivigitee by B= Qx3p; =ax3! =ax6 =lo. TE He lees of the ued MASTER ane MOYB.ond the locas the. foomed coe ona once, cthey find the vonks, of the ued Peorauted In all peesite Ngee in the lictinnas: Gh REMASH Ui Piree’ ty MASTER 1 1 st The Olprabeticad ovdes of the Weed MASTER iS AE M Rot othe begin vith An ara a f begin withe —-—- s--8} ” Povipery begin with MAE — — — =31 begin wlth MAR— — 7 =3), beginwith MASE — — begin with MASR- 3 beg) MASTERthe vonk of the wood MASTER=5145)43)43 +9349). Y el . =IBO*IBOF 66+ 94.944 i =a5). dW) ReMAsT the ‘alphotetical ootes of the wood REMAST is cM pei the begin with ALL --- 451 begin withé = - — gthiaith™) — ena AD begituihRe A — — — fag pena At =o! the Bonk oF the wend REMAST = 2614S 14S] FU/43!40) : 2 04190 +180 F9U46H ; = 3a). 6-| TE the leHews of the wooed BRING ave peomuted inall pose, ways, ond the woeds' thus, feared ode onsarged in tesatceatey Joodes, then Bind “the Sots uxed: D>] The alpabetical .osdes of the woich BRING IS‘BGT NR ny the beg with B- — — — = =U begin with G —_ — + > begin with Ta begin’ with TRGB — ~ begin with 7Gy — — TGRBN + Sqth word = 2qReN. it Wd : Thee 108E 9 aeeed 9 koi Out OF 9 bjs, Seonrict A [in thee Sera), tones’ tei mony Goange nen Con be rode Such ‘thot each object con be putinene bon ony” r-| Since 5 objets. cannot fitin 3 ‘sma: BOWE « 5 objects con be Filled in 6 Srroll-/boves "2 Remaining y abjecS ao be ossonged DeMaining’ ay ways- the tote) Norof Requitaed BY grange Ske xat | Ixus Gp, mys 4 bores 9 . 2.0 Yas | mm hore, ye 24een gS = Me See cu pio ae pi! Ina SHE | oot, seach ied ey Sous foored cae ossongal 19 dictonery at PFaNK of the ocd PRISON: J ate alphabetical osdes of the coond parison 16 INOPRS: #') pe begin with po begin with NL bain with O - ~~ ~~ 26) begin with pr- — _ _ me begin with PN~ — —- =u) begin with PO- — -- =u) bain with PRIN-- = 2! begin wlth PRIo-— - =) ’ begin with PRISN- -=1! PRISON = =ol zo the Ronk of the wootd PRISON =S!+S!/+5) +4! TU! +4 Bl +8/+t1 +0! i =1R0+ 1DOt Wo+Sytay+SYt At At I+) 360*718 = 438 Tf the lektess of wos CAMCET Owe peortitel In alll POSsiblo woys ond, the woads thus fosned axe Qesorged tn the dictlonosy onder, find the vank of the uosd eAM CET. Sl]The alphabetic) oxdes of the wad ¢ 2 AMCET } tie gin with A= sg 1 Bic GCE: _ bain with ¢—---- 8 t ; a bein with € ae. =3) begn with €AE--- = 31 begin with EAMCET =o! The Ronk ‘OnK Of the Loosd. CAMCET= GZisSl 131421 +0) = 1804 12 “get BP +6 +641 = Bug!?0 eo +6241 nis Ut = }26412+) > Votl3 = = 133—~ | oes] Pesmutoytions Lohen, ceptti tions. are lallowsed- let hand's be positive integeas: af the, vepetition of th RS AMOwed | “then the numbes of peomutations of ny’ dissimiloxs thires token” y "at a time te n® ; : AISTE> the number of Pesmutations of rm, dissirnilos thi toe i things. ota time with atlost ene sepetition s Po\odzonse = A numbes os 0 Loosd sohich seod8. Same ithe, ea: Foon lePE to Bight Ge fear digest to leFE IS aniled Palindcome, AMM, ATTA, ROTOR, 193915 1a00a) etc. te MIE :- the No.of Ninckorres, fore bs Lente with distinct’ letias’ thot aan be USING given n ‘distinct letlest 16. npat iF ow IS een. | - 3 t 3, OF IF w re ald - Bc tae Exeacsse 5 b> Ts). || Rind the nuke of: U- digit” numbes that an be Bomed using Ahe digs Fau, S/1L®B when vepetition 1S alowed: 591+] Given Sle, us, 18 ee i oe “LS 6xEKEXE* H : “= 1896. Pie ; Find the Purbes of 5° lettes usoxds thatan be ened using the lettes of the cocsd RUYME, if each le Hes Cony: be! used ong numbea of tinnes- ; SO!) Given caced RHYME z : hee ras > ties ‘ wB ! > Stil tr ry mle- of S lettes wood tS ness = S¥SKSKSXS = 31as. SB || Find the humntes of functions foun a Set A conto, ining s elements into a’ Set & containing Y elanenis:poof elerrents. in A= Neem ere? norof emenis InBe Keaseg > i nay? crn oo’ 2 neof Functions fon Ao Bis NID 2u> 2 = AKUXUXUXY ' = toau. = Fai] pind the pumbes of U—cigit telephone oundcecs that can be Vlfgpmed! Using: the FigtS 18,3.4,5,6 with atkast one digit 2 pepentee et} Gven digits 12,3 45.6 7 n=6 \ : : vey: fe ThE HUMbes of Y-di git telophone numsbess volth at least one digit vepeated= ne. 4, é Ps. wel Sr, 5 = 926. "971 eles 3 |Find the numbes of onto itser ep} No- of elewentS in An NCARF t No-oF bivections fons A to A IS NHI Bijecttors foors a Set 4 containi at! ey so souo- - W | pind sthe numbes of cays of awvoting things, Ina line Ussing the Qivenn’ diffesent thirgs, in uhich atleast ane *t ing iS sepeateq, 2) the nunrbes ‘of WOYS OF asBaNgiry So’ 44 Quen’ diffezent things in ushich rnp. IN © bine using the atleast one thing iS vepeated’s S | Find the numbes of 5 lettes’ woods that can be feansad | nq the a pleted Of the .woged, AATORE that. begin voith Nude, Zepetition is allowed: mL I S9\-| inst, place we confilled with N th one voay Bemaining “4 places, can be ftlled with ongene of the '6 letlews in Gp, xOpxX6px6p, ot rabes of Sleltes woud begin Leith x = 1¥6x6x6% 6s Cth206. Find the nurrkrs of S=digit nurches, divisibe by Strat canbe = go erred vsingthe digits 0.19,34,S Uben eepetivion + Fist pace must be Fited \ Lolth, es \ = ©2 ©) BOS UOWS iN Sp, ways, lanes ms Satlooey,lost place must be Filled with: 0057S In Sp’ cooys. ody of the othes places can be filled with Gp, warps. 6p,x6P:x6p, wage, the % No: of 5 digits! ndinkes 1S divisible by 5= Sp x 2px SPs 5, = SXAK6x OxE = \220-2160 =6 [Find the numbes of warp of SSO QING 6 koe ond | 6 give in 1 nha eels a hous mor oP bese oscar ols 7 Ball the gids ose togethes. tiyno t2o gisks ase togettrs. & bes ond gils ‘come alteonately? alt 8 bop+ 6 gisis= 18 peoson. they can be peony INOS! ways. et 6gtd> 03 ae unit: OP then toto) No-of pesons= 6} + tunit =1 ways: Then, the 6 gids arrong themcetves, con be pesmuted in 6) Ways: the totad No.of awangerrens all ‘the Gls axe togethes = 7) x6) ee | ae ane togethen sar GhaS let, 6 boySTaBeanged i in 61 Ways. the gids con be ossanged in the — gaps between) the eee in below. yl UB XB x (Bix (BY 6 Nous, vege, sable aasorgia io 2° gars Te HEI TS othe total no-of armangennernt nO Spa ee # BYR BE * 5 2 ry ¢ S614! nn =1x61x6! ui) boys. ond_gids, come attenattelys | (ght 7 eS [tet us toke ta’ plows + Thee moy “begs” wh ey % yea woys: 16 Bou, canbe essanged es 6) Looys: 2 6 Gols con be songed in 61 cays fe the totad no-oF cmsongrrenis bap and gals aoe Oatteanatery= akep eG) OC 8 aah walleeanr ats oeitst *eftind the oumbes sof UH rettes wosds Hee Bebe Reina essing 05") the lettess of the wos MIRACLE. -+tlow mony oF ‘them. dbbegn with on vowel dir begin and end with Vows aia end Uh a congonant? »|the wood MIRACLE hos, 5 letters. Henie the nunckes of uletten words that con be fesmed’ using these elles } 1S Tp y= TXB XS XY dy begin votth on Vowsed:- CI — post, place must be f warp ond semainin Letter in 8p, 2 roof ye Yo. ‘led with « one of 3B Your's A, inap, 732 Places noust be Filted coith 6 em ainirg letteo voonds sthat begin with & vouoel = SPX Ep, Gs begin ord and _erd_usith youete: [apace prost & lost place must be filled with | vouwes ing Pa MOY: Remaining @ ploces, must be filled usith cernaining ss le. Tet 19, Sp, eays: Norof Yy- let{es wooeds that begin & end usith eee =p.x Sp, = Gx20= 180. gid End vith _& conspnant:— toot uot place “rusk BE > Filed with one of the ore rR ois Up| WAYs-9>4 We bay 16 Remaining | 3 “places aust be! Filled sith vetrinig Bileties in e Spy uagss “ y 2 Not of is telked _voceds, that end orth ov coreaant = Ae . ‘i ; : 1 = URIDO=485, | Find the ;numbes of wooys of Peomuteting the lettees of the vy [wood picTORE 0 thot SP | yait Vousels “aenne 4ogethes’ tisno-tuco: vowels; came diirthe relative pocitiors, of vowels ond romania aie motdstotge By. foe wood picctoRe hes 2yourls GOs ond q consonanls¢ Pe. CTR) & All vouels_ cone togetheo Coreides 3 vousels ag one uni: then ue can assange Uconsonants+ Tantt of vowelsan Sl ways Aous the Svowes omeng -thern selves © can be peomuted 19 3t Ways: a |the toto) FS: of Esp totions on vourel, come “egethes 1 ; PO By BIS 9 b ' aie ee t = 130%6, dh No 1000 Vousels cena 2 es et epee iene: feb, Y aneonants can be ase In, 4 LEays: Then in betcocen the vowels 5 in the begioning. and in. the seating grtreve,cwe Pre as on betous % r a 5 ih tee al a 3 xy el o Ses ts, 5p eearpe * Totad “Ao-of! omergerent no two Vows came @ tegethen yx Sp3, | = UNWSxXur? ‘ > =~ FBUNSO, fre feo cies cit tlio the welative positions of wowela” ord conseqant=, one not N oO]? distuabed. ; a nost | the thee vousels, con be a ad in By and denainit | St coreorents’ aan be orsanged In) uyecags, vieMc | A AOLED nto-oP emnengensents the celedtive ions oF vowels, naa Gencorants ase not distrebed =21KU) 2) 4 S Bxaq ' P9160. sod. LSS vias = 23-5 aA hens mar ways 4 nestosan ta Popes asibe <0 Meat the 4 cond! the teat “ay prey Come shes db) May not come shen) = the best ard ‘tucect Perpens, ace thentel as. ene unit, then arsine “Hi = S papeas- Nos these’ can he in G41)! i-e Srp ua WB and the beatend wast papas betioean -imncelves can, be onsorged in a! wore ith iano i9 ayee RK att bog stan ate eM asnbriirestsThalg Shite tagethel |S) q sow 5) Gir Total lnocof, cooys Of emmahing mathernattaal papessaisigys pte - best, amd occas: (POPES cone rtogethe. IBS MN asoyp - "Fetal No.of agrangements rray-not. came, ; togethenz Q1-8)ai = Bica-a)B “pa Ne ger 3 z ‘8 "the No-of ©f numbers leas thon Sano - U+16. pind the mumbes of patindres, with 6 digits that con be foorned Using the digits, ¢ %O8U46-8 13, 54.9 the AB-OF palindzomes foomed = Up xl? = Uys 2 = =uxe™ = Uxd9S 100 qre No-of polinckorres, fosmed = no _ =e Ves 5h . = 5? on 9 ¢ : 218s. i Find the nurnbes of numbes las than Scoo that can be fosred using the digits, 1,9,3,u if sepetition 1S allowed: “AU singe digit numbeas , tu0o. digit numbess_ theee digit nurkes ord fous digit numbes stosted ith 1 jase, the numbess, tess thon Seco, _ bathe) rlo-of Singe digit cubes = up, fothe no-of teoo dig numbess= Up,* Up . 7 z a ily The No- ot theeé digtt Nunn bess = Up,x Up x Up =43 ee 1 Wothe No-of fous digit nunbes stasted eth (= UK, px Uppempes? > 6utéey = (yg. i 9 diffesent lettess ofan alphabet ace Qi ven: Find the Dumbes of Ublettes Woceds that con be fered usirg-these A leties UObich rove 4) ro lettes is vepaxted “ti atleast, one. tetten tg vepecited. “| No- of letters (n= 9 Given oe 4 : s & Ro of ulettees uscd that can be foamed byusing the Rivera lettess is PR, = Ry = UxX8KIKG . = 3o0ay. duatleast one letter ic epearted )— Ao-of uy letbes med -that Con be —focmed by Using “he Aven o letters with atleast or lettes is vepeciat “|8 np, = ft Op, = 6561— 20ay + 1 =abet the Fisst’ poce ‘can be Filled waite 13/3, 2 Find the humbes of U-digit Numbeas thatcon be -oomed using ~ the digits ©. 3, 5,73 that ase civisible by dauby when sepetitin, is allowed. 5 Given digits 35 18 2 :- Ww" Piwst place filled with a}js/4/3 in Up woys last place Fitted with 0/8 ]8 in 3p, ways: Ramaliting "g places con be Filled voith zeme, ony one of the quen & digits In SpxSp . f The mo: of y digit iS -dtvesible by a= Up, ¥ Bp, * Sp,*Sp, FURXZBMSASH boy di Dwisibie by aie 300 ( 7 too FRSt place con te filled with 915/4/8 Inup woys lost, places Hilled votths cody 08 08 20 cv 28 ds 72 06 S2 On BS 03.80 In & woays. Rerraining a” place filled with ony one OF the 5 digits In B ways-7 ae . ; ; ee, rica U4 digit IS divisible by U= Up xBp,xSp; rg | Find the numbeo of d-digit humbess, +hat can be focmed Usi the digits 021-2 3,4,5 wudhich ave divisible by 6 hen vepetition of the digits is llores , - pe i jj 1a/S Ins, | iael poder cot Be BU Hh. oe PAS SSP eaoysoinent novo, the last place can’ be) filled? ‘ustth only one eoay + sethe totad rlovof Reqeioed ompangensens = Bp, x36) in =SX26x) : i =180 siiBP lrctes if Pag wean ad o} ma the numbes cf ccules peomutations fn’ dissimiles, things Haken all ata time) ig gy). . We asthe numbers of ciscules peomuitations (Polen of Flowers, ~ +f Fein oF Beads] of n things Loken oo ata tim’) is sin=0! 4 be Ereacise 5c) ind the pumke of a | ee ‘ays of onedlng 7 paxona oucund o ciecle 1S Cn-t)! ® of omongeg. 2 eee the numbes of coos of SBBANAING —F pessons around a ciocle ig G-pI= 6! A we i i =730- %[picd the Numsbes of say of oxzanging /the chief Iministed Ond 10 Cabinet ministers ot a: croculs vlobe: Go trot the iS apietce olay Sis ina posticulas Seat: + Bri eAurbe Of pexcos =1 the NUmbes of Requived Orrongemenis=1 G0 | ° 3 | Find the murcbes of ways OF poeposing 0 chain with 6 ‘cliffecent colcused beads» We Kroes thal the nurckesMoays of paefaring a: chain. isith"n" different colcised Beads btn. : . the number? ways Of peepoving a chain with “6 “cli'ffesent Celeused bende: (6-0) = SI _ 180 - eo. Tvl Find the numbes of woays of amssanging u boys and 3 Qala, omound a ciacle So thot ait the QisB cit “ogethes. : € £91712) gals: teat 03 ‘one Unit Sematning | U bop one UUnis « Sunits con be owsanged Ina ciocte 1S GOI U |= af. this gists can be ccsorged themselves, in 3) ways - ?) The totad No-of seq uiced exvangemrentS = du x34 [Find +the mMumibes of voays of osnonging T gets and Yy ladies cxound 0 cizc) table IE no 4u05 Todiexw Wish to St togethes: _ . =aThe 4 ladies con be expenged 3M the 71 gaps In Tp wos, & The YentS abound 6 | cocates table 19 G-DLe6y, { “the nos of vequised cmrangenentS = py X6) .” ‘ = ae >a the numbes of ways of eeeonging 7 quests ond 4 heat > & fosound a clscle HF @ pastifelos Quels) wish “to sit on eithes sicle of the host. Ve boy SOl4\ the Alo of guets=7. The tio panticules, quests aleng 9) hestias, one , unit TIS six UNS con be emonged ino cwattlas tables G-})) WAYS. This tL20 guests ORANG Mang, therselves, in B! WAYS + SAME NO-OF sequised amengerrentS= 5] xa! = 80. ae Ue) | Find the! numbes of Ways of assanging + guests ond ‘al Hest Gsound a dacte\j F-2. Featiculos quests “uth 13 Find the nuinnbes OF Loads Cf peepasing, O gpaland colth, aijellece UWeohite ond 3 sect voces’ oF diffezent Sizes, Such that the tia, ~ EL CEES Cone thee. “ i DI] Given pile voses =3. ats Givenitohite, poses = Y. : 5 J 1 adh bo Given sed v0Sa,=Q . i \'| Tex0 wed coses, a one unit, semaining = soses 08 + Linits- THIS Banits caouwd a gqeserd Space 1S £ln-1 1 =5@-9): Te00 sed soe, SIR, cthereelves, in kee s1 eoad, the Alo-of mAQUIned sevargement FlB-191 KB! rata WED xa! a i : = Brig = S00 : | paisa : a Et: | Find the mumbs of pedmutotions of U-digtt nunkes, that’ cin “The foored using the Agits 12,3,4,5,6, cohen Tepctition Is | ctlowed : ay : t PW Given’ digits 19,3 4, S06 - n= No-of 4 digit Numbes = ns i - 7 aa bre Beep rm Pupriasa lg 4 os uyDPT picd the nurkes of q lettes = lea the letters Of the tcosd PISTON! In uhich atleast one lebtes is sok gal Gheq tend PISO! H hes 6 lebteas. the aunebes oft Ieee | Hence», tre cmon Oi0f unsuccems ful attempts Woods thot con be foamed 4 oepeated: wots thot mn be foomed using these 6 leties. do when vepetiticn is allowed =mPe GY gerhen vepetition Is rot Aowed is x, poe dit tn Hence the nuirbes 4 teties usmle in? edfth atleast ene letter is -sepeated 1S ~~ ' . 4 oe MPs = 64 & = 1aa6-360= 936. : A nursbes lock and 3 “sir and exch ai ho 4 digits 1,2, q+ Find the montinnunms numbex of, wnsuccesbul atte my thak Con be made lby a Penson oho tries “to open tide lock Usfthout Jenovsrg« the Key Gde . 7 fach ving can be yobokd tn Adtfsxent woys - Thus, the +0tal now dthperennt conys tn ishidh oyings can be-votaked “2s 9? Oak of, these atternpts » only one atternpt is successful akternpt- Sit wR 2429.4) = 428 ilfind the moiof Ty6 ith 7 (eter pallnchores Erot can be a ol)-Cfven word EQUATION , 5 &: Fead the mumber o| formed uxtng the Uetters of the word EQUATION n=s ‘ nts ‘iy Novol © lektex palindrome, that can be Po fe lette’s of. EQUATION = in¥ = gf sg? Wr Novo} 4 tetter palinchomes thal can be formed Lote Le cles! of EQORTION = TES git _ oH 7 7 formed using the | Severn digfted -palindiomes that can be formed using D,2,%, baal a 3 Fest place can be fed fo 4 voays (using only non - 2010 dfgels>- Remaining three, playes (an be: piled % Suodyseads © 5 Noxo} :polind- tomes = 46> . i cod anaesPus) obs os TI i-| Find” ‘the numbes of coos 2 Sko i of eesangl Ged cwoees) and Soak, Des —of-diffenemt—aized! accund & Ccalin lable oe far Dall the gisl> sit together \diynotuo ¢ QisIS Sit! togetnes WO bogs and givls sit altesnately. ‘ = Sor-| d_atl the gale sit togethes ve ' cr Sal All the Sint gidisias ene unit 6 bays, as) 6 units! they can be omncinged tHE Tunis vourd a cieculan table ing—b! wong + ‘ jo" GS gqiels cmmarged .amorg therrectves in 6] coays: Th . 1 =O-19) %6} e totad oe of vequived PET ee ies ia No tw gisls sit tegethea:- the 6 gies con be apzonged Sp, ays the Sboys omzarging @zcund IO. cizcul table in G-! coage. & oy + the no: of aequized eavangementS= 6! x5! (Sr. XG13) dh Bop and grote, sit alternately~ , Prost ossarge the B gis exdurd a clectilas table in G5! won, Abe 6 boys can be aamanged in Se Oye seothe totad Aoof neque donor gemets= 6-19) 6p. Si x6! aa Re EE oer Find the rlurches of ays of aseargi ing: Sed waSBr and 3 Yellow poses of differnt sizes into a gasionds Sn hao rrony of then: dy all the Hous Tes Ome logetfed tino ‘Loo Yellous boseaane tegpthes. 1 Totad No‘of | 6s =Q , : bes of ways of amanging Gxed'zoses, ard 3 Yyelloto Doses, a gee ee oes 4B! = 4OR20 — BOIbO- . a 2 yellow 2aSe3 a& me unit semoining @v0s63, OSEuNitSy his 7 unit con be orearged: qooland. Space LCD, toays: | ’ Bylo cosea omranging among tHemselvel’B) Loaye,, w the Morof sequised Ommangerrents = LO- OTx3) $6) x3) B 2160." 8 La a a i A chain of beads 1S to be poeponed using 6 diffeserit zed coloused o Awo c . - dy n° Yelloco zoses ose togethes- : ’ Ph ‘ OUD “OSES can be asaonged IA 6 Ae, od ashy OF “ ee ate } Qeslord space in 4 (6-D} WOE’ ' athe Ko: OF segdiod o s : BAER comorge en = 6. «1 C60 lea =I0x Lao) Se hey j— = 1800. ‘ss A vound table cofeserce is atterciee by 3 trdions. achinaé®, Scaradias , fos ® Ameoicons: Fird the uso Mumbes of coa of aen0nging trem at the zourd tokle SO that the delle, beter min Game countay Si Hi ia Qe in oy gince, delegates, belonging to Sanne countoy sit togethes, ae Foot, osvorging she form countoie, in azound a cleculas table s GD! ways: lakwee) B IrdianS con be exmmonged among therreslve> in By eoays: @ chirese can be asn0nged orerg thereelves in BE wOOtE: & conadins con be assanged omong thearcelves in 3) vongs: Q_ Amesicons’ can be Aoserga] among thenselves IND! anys. sthe No of vequised ormangrmnis= GED INS XBI XSI XA! : GX6KEXEXD - 5A beads ard 2 diffecent blue coloum beads "an how mony cays @nthis, ve core sOtrat no tino blue Cobims bands come togethes. ‘the theee blue colouced beads aan be ansanged in 6 gaps N6p, waxes 6 ved colorméd beads in $(6-D) Ways: “the Aovof sequised osvongerent= Ep, x1 C619! = 180% 4.0r0) =a00. A family consists oF fothes, mathen, @ doughiess ond 2 SereTo 5 hou diffecent coos ton they sit at Oo sourd table Tf the #50 cx Keo dseh to sit on eithes Side of the fathes. : ; the tetod Noo pecsans in a fomily~6: : . 8 daughtes> along wth fates 0% Wine mothes,., 9S0nS aS aunts tS Coury UNS con be aczorged ewourd & coculea table is 4-9} & The a daughles aan be ocmanged eithes Side’ of Fatheo 14 31 28 [fe the Ao-of cequioed acvangements = 34% 9} \ 6x2 \ t2~ eens' —— 7 it ere bes of Li ti of a\o" ie nuns of lineaxs peoroutations Yi thi in udhich “A +thy ~ fore a.fke ond the vest: ave diffecart isl “em Rbep The Rumbes of tren eh iN Ohich these i , Kind q fike thi of Eecond kind Blike things of “the thind ‘kind ond “the vest ‘ase diffesont ic Exes cise~5 ad» Find -the nunsbeo! of ways, off omsarg ing “ER betes! Of the Loong, @) TNOE PENDENCE : . ! ee oi: Given feiss, TADERENDENTE alo-of letteas= 19 Noro “AL "3 = 35 4 Novof “DY Df ee Mea xd cpr ran | Hor. okt EM eordt y . yy cowie Motel Ao-of | Assorgement5=: 18! \ na eee . - Bparar LPLaYsy ‘ db MATHEMATICS aoe SOI> Given wood EM ATH EMATICS Alo of tettens,, ears 1 bead do oiorb Ao | Novof, “> ‘ 5 b & bie o& \ rid OE AAS! 999) ebrtcd ytd aaah on Jos 4 , Roof S'S be vaon ot hast i 1 ovat [rotad Alor oF Aawangedranise! MN ot : . ayaa | diy SINGING: lel Given wooed “SINGtAG No-of tettesS = F.5 No.of “rr = 9 bes Alo, of ait?) = 2) . Ho port b yoo o Ao-of “Ga Bash ak bo ot 3 a ipl Totad ato. of eligatl f aya) el ‘ . Qe COMBINATION® . Wiss grate SQ1> Given to0sd WMBINATION + i oeB ya EATER MEDI ATE lect Given wooed INTERMEDIATE Mgrok REOS= 1g | por of HS = 5 orok 422) 9 . Norof “E'S = 3% : soe! f Ap TeRED Avo! Aosongenente gate iv) PERMOTATIN | ys Given word PERMOTATION No-of lettess= u ‘ Novof “r"S 28 hotod ato- of Arsongprrents - a . es) [pind the numbes of a-digit numbess that con be feomed cog | 2, 8,% 3,3 Hy, Gwen ckgits 9,23) a ar uy No -oF alegits =9 dorof “ar 23 No-of “3"2 = 9 No-of “gS = a [the Hoof 4 digit numbeoS 8 sing 3. ]pind the nunbes OF Unletles: woods that @n be Posmed wine letteds OF the R ccood’ RAMANA eo) Given wosd RAMANA No:of leties = 6 Noof “A> = 3B - novo, woe Find’ 4 letters Looe fea “the Felovsing cos: coset Any letter diFfecent Ghe elovof O) ithe, nto Of U, letters voosd, foomed YH] -+74 coset: texo eare and noo diffesent a the rlorof voces foomed with RMAA> Zb= to the Noro wosds foormed with RNAA= GL . , the Nor of Locsds fosmmed usith MAAS GI = | 2 2 26 Casedys 3 Some one diffeserrt the nlo- of coonds feamed swith RAAA the Norot Loosds, fosomed uth AAA Phe NO-of Lootds, -fSumed with NAAA 6A 12Sol Sot’ oT] Hou many numbess con be foored using all the digits, ' L9,3.4 S32! euch that “even aie ony evn Perce Qiven digits £2,3.4, 3,2 Ithese ase t100"I’S, tu90" ‘oS foe's"? one“u® thaee even places, can be occupied by the even digits a y a a ways * The zemaining edd places, con be cecupied ‘by the odd _dighs L3,3,0 to . ayo) OP ethe alo-of sequised oxmangments = = BL x Th a liboooy, these ove 6 Copies of one .Ea0kK, Y PIE each of tco diFfesenh booka, 5 copia cach of thee di Ebeserst Looks and 3 cepies each of tuo ‘dittemerk books: Find the nuncbes Of Loays af jasnerging all these books Ina creif ina Single wu f the total No-of books ina Wsoerye 6X14 UALS SYS ABAD 18 64841546 7 = 3s Toto) 0-of vequised ceo ents'= BS 1 at v rence eny soy] | woe NNTB: um 2017 A book Steve hos ns, apes each. of Sn‘ diffecerst books: FInd) the nurrbee of cangs, of cavarging these books tna ele inasirgerro athe no-of beokS in a stese = maAmxm ----- nllimes) = (on) | ey Lhe totes? No-of vequised pak oon mm Find the nunches Of S-digtt numbees, that con be fesrned uaing the digits o ly ly 2) 3- " Gen digits Of L383 FIeSE place con be: Filled _osith ioo oD B03 in > Up uanys The semen gy aig- Places con be Fite (gith semaining fous * i: “ithe Aloo -scguived cntorgerrent ~ Up a UK ae Qo _ ust 7 rlDPTan tee many WaYsi Gan the vtetieas: ofthe Lod cleeSe be 6 | rerged S© that ro two E'S core together} Gren oad CHEESE I gt Ro OF letteas = 6 " “novof E"S = 3B wol wil sy P prek USe ORBAN EE Cy HAS In BI uays Loe hove u gapsinzetuccen thee le Hes NOU, WWE @enange 3%e"Fin U gaps othe ways Hthe Roof wayuiced cmnangementS = 31x bal 23x he ar = 6x zy @ . = dus = By. [pind the numbes of ways, of cxeanging the letteas of the Locnd cape ASEOCIATIONS, : T™ how ony of them * ' dy au the SS"> Come toge thes isthe too A'S do nat came togethes® Blt] Given oad ASSOCIATIONS: No: of letteaS= 1D Novof A'S = 8 noofe"? 9 = 3 d Nos of o> 2 No: of E> a is 4OtOd No-of vcequized assongemets = am Al, the , thesee “S"S come togethes = a 7 n Can , » YO \ ses 4 Ade IATION Lunt + Funts = tounitSs = 0} eae aia, 3) C ~ Coreces ' 3°S*F 03, ne unit semaining 9. lettess 03.9 Cerise this © ens con be osvorged in, 10! Bla! B! 3°3"" con be omorged arerg -themce ves, in 3) nys= Vesey. \ i [the totadsequived ommnrgerrents = aes"! , ‘ AT sr ( ; Ss Sh = 3)is the. tao A"F do_not come: -logethien’~ i LSISL8 [eu TITEL OAL Sy Fret, we asvange all lettess except 2 4"S in _10} 3733)” betvoeen to letteos woe have " gaps in this 0 gars we cory Jensocged - @ ASS = "Pa woop. tothe ‘totol Alo of dequi 5 = 10! “Ry | of vequiaed Secrprert Spree” 2 " @ | pind the numsbes of ceays of ossanging the lelfesS of the — econ i MISSING So the two's! Sase togethes and the 40 TS ose toge thes. Gen wood MISSING. No: of te tteas =F . nNo-of “ms 2 rr" . 1 No-of “s'S Lg \ : i ne thy lett eet Buns a + MNG. = 5Bunits b eA a mt Consides B'xSas ene unit ond BS" as stunt veruiniry > leHeS aS ZB unils ths 5 .unils conbe ‘omorped BT"? con be arssorged orong themsewes, In a"5"> con be exeonged ermong cther-eefves, 1, Fthe totod ato: Sor: in 5) Woy. Bh ways. a “of sequined omorgements'= 5 | re * “ar a &£ post SHwiK ) de> TF the leftews of fhe woad ATANTH ane permuted tn/all possible weoye ond the coonds thus fésmed Ose orssonged in dictionowy ovdes, find the Penks of the umsdad) ATANTA Ui TANATA. sor+} db ASANTA- 5 i Given voosd ATANTA - the odphabetical codes. of the wood, ATANTA is |A AATNT fhe begin with Aa — — ~~ 2 Rh le beginwith pA ---- 4 i 2 “Begin voith AZ AAL LOLS! AA ses ure 2y begin’ with ATA N Arrant 4 begin with ATANTA =o us athe sank of the Wood ATANTA = H+ Sr talty) |1% eat 2 WOrlat ari ~ BE Ww TANATA Given vwooad TA NATA the odphabeticad cades of the wd JANATA 19 AA ATAT "DA prart Ar Kt the begin Lolth i is eet Ao! -2 =Sh 260 begin, with A= = = — ~ = Bh Mog begin with Aba = ee -ai JA A= ae begin with, TA A -a- = Bh app an ait =) 2) 2\ _ begin vith TA A =--- = i Lp 1 Bi NATA= 2.6 3 begin with TANAAD petri , ee begin with TA NATTA =0),° she wank of the Word TANATA=5!,5!1 , 5! 4 Bl 43) 41 3! 3) St oO! ho = 12079 pe? 196? 6-3 Me TB + Be EASA i+) = 604642. 6048 =68.. COMB IAL ATION ?- Combination means only selection of -things. ‘ the numbes of combinottions of ry diffesent) things’ token *s' ata time ig denoted i by Te, ond is defined 0% Peym Oh os) NoTe:- Holo) b the numbes of diffesent subeels of <3’ elevmis ofa set eentotring ~n’ eleleents 15 Nc, J IF n,~ ave integas coith oewen. thele= MC, as | o = 3) the numbes of woys of dividing ‘mn’ dissimilox things snto Wr equod poy poi aontaining in Velernents IS. 50.01 wy, mol) |G.the numbes of ways of disteibuti Hbiege, equally oer. mr’ perso "S ny apm] 5. the numbes of diagonals In an n- sided Polygon > Yon — mn? * dlissimilers G Fos Of 0, 04n if Ege ey ten . ie the ; a EE leer Da eR dt IS. the numbes of ways of Selectirg one’ os rice! = things out of the given S19” things is gm. Exeacise cee 5) | Bet} et Eee ‘810 findn. ‘con Beg 23107 = 7 Sano 5 lomx | = OK GADXBX7 3xB = (OKA BK] 7 a Ur Dkr) on) ’ aq, = tO, n8s| PF Peg uas Find the Peels alues oF = : exit : 1aXQ (2 XX \OKQ ~RAXSK AKT 12 > 120, ca “elu 122 |BOUIO = 56 *Ax10 = 8x8 xax10 . i SlOXAX BAA : = [ue ” Te ~ 4Sone Mes, 5 te cy then BES - Be+l= St-S | Vs Ot+14 34-5 7 d= 5t-y =5t = =F Bl em bait = 37 bo 0 saat al } 4 Sqy>| TE 88 =!Poys then Stata. %rTa_g = 9 _ 4c, 4% “e = Ss nner et Pe, Fat I ey % Sa 4D, = Oc, 2 a log) = te, Qo lg = Cong ep) = : 'Qoe te= 2 3 ma 2=4o» @- Cg "as abotd T= a fo. _ 1 a) a gr Ss 6 Find the numbes of woays of fering QQ Gmimi tte Obsmerubes, Foon 6 men and 2 toxics. Ao-of men=6 Me-of tardies =3 The toted cto-of Peeson =q = “Re clo of SF ot fooming 0, Jrongmirttee - ef smenibeas is a} ax wx XOKEKMI bE: Fog = es ¢ = Gi S1), GBP aeucston nor F = m aU OO, FOLD many Committees’ contain ae | i] pcoo tad norof men =6 ‘ B'[ ion oF todieo= > : i 7otod Nov of pessons =q Now, we Select O Commitice of 5 membres vith atleast 2 lode + mente [ORS Gy ersten | 2 9 Be, « 3c, So 3 6 a GS Bey NoreF commi tteep= (S,% 3g)+ (Sex 3a.) = 20xs + 15%) = 6041S , = as: wa] DEM = Pg, then Find Cats 5 Fs = 13- Tote poove that foo 34 6£n, Cav] (ep, « 8 Neg BO ree pO =, Spe Oa + BOS. nyt BOM O35, Feat "sy" ), BSC! J SOS! LY t e ” v2 = &2 ‘ al oS oH “oot Sy - , ; AOD, 4 OR 4 m2= Mam Sent : Pate % Seo = m4 Se =. Mey ® | pind the value of eae Ee + OS epir| Weer B. 1c + . Z : = es 1c, Woe eo. = a 4 a = HD, OMe + lores ( Regt Pea 21 of Gaal Gaia '0%5 =20., |agenals ofa, Polygon eoith. /2.Sidep, : Find the _numbea of ways of Selecting 3 glals and. 3 boys Ott ¢ oat |pled the nurcbes OF, di E00] No.of Sides (H)= 12 NOL OF diagonals, go 8) eco & Cpe BORD, Bray | oy, Fn -PeosONS awe Shing in Gcou, find the manus ol lee Selecting tan Peecons, abe Ove Sitting Wocene to Snare