MA 102 ExtraWorksheet on CountingProblems Name Showyour work!!! {-- " ..1.,..i:.."];.;;..:,.':.1'..: 1. The board of directors of a corporation has 15 members. ln how many ways can an executive oommittee of 4 members, (President,VP, Treasurer, and Secretary)be chosen? 2. How many ways can the sameboard of directors choosea delegationof 4 mernberswhere all the 3. There are 20 singersauditioning for a musical.The director wantsto choosetwo peopleto sing a duet and everyoneauditioning is capableof singing eitherpart. ln this situation: would be the numberof ways to o ,.r&.re, fto I t ,oC, C-lr,: le +,', o p erf t-L cr.,..,.o,-.* fo F,.r \ O r* rb acs;6y,4 would be the numberof ways to l).wos<- ,oP, J-^.ro WUd-u ^,"-^,-\ 9.a ,,tlro r,< . , e-or2V\ ;s, dss . r&^e! +.. ^ d,fel'.*^* f ar*. 4. How many ways can the director choose a lead singer and a backup singer frorn the 20 persons auditioni"gl .,fr\ C Yaf ?-o & to r)l 5. In the situation where the director is choosing a quintet (5 singers): ,o{ would be the number of ways to o! 5, { f 5 c^/r(\r\.\ t }qVtor, le rpqf +5 ql @/u ,oC, would be the number of ways to 9-D Q.^r) bz- esgn: 6. rhereare r 17Division r-A co'ese rootbail'ffim9.'fr(Y,")";'.#,#t3i'f#;l;*g V*Ao 'anKlngs possible? tal ll'a ,'I ,L t I | A t]t. - or -dtt ,aa'-t' [r, -;rl I I ^ 11 x l-f orrtL,v, t17 Lel-. 7 . For the same I 17 teams,how many ways are there to choose8 teamsfor a playoff? tftc-, 8. Four mon and 4 womenline up at the checkoutstand in a grocery store. ln how many ways can they line up? ? = e7k.5 8'e) ttl- L"a 3* 4+ To bo K.c {rrrr'., 9. In how many ways can 4 men and 4 women line up if they must altemate woman, man, woman, man, and so on and if a woman must always be first in line? -D '-=-;A'a'1',+-: ? -L"1.:'-=w r\^ 4f4'+l+ w ^/. r^ W fyl w uootnxe.*- 4 rwn"'-- 10. The ski club has35 members(15 fernalesand 20 males). a. How many ways are there to choose a president, vice-president,and treasurer(no one can serve in two offices at the sametime). - - - = ? 3; 34E I?1",J7o*"d 15'3 b. How many different ways can the president, vice-president, and treasr.rerbe chosenif there is the additional requirement that the president must be fernale? . 34" 22 t5 lc,a3 o-o 7 vY T c. How manydifferent ways can these 3 officesbe filled if thereis a regulationthat says the top 11. How many 7-diglt numbers(i.e, numbersbetween 1,000,000 ar 1 k_&_ /o /o /o , *rs DOpoo r\qv^\onrs o-1 o-q oJ( t-q o-a & Tur,^_ 12. How many l-digltnumbers (seeabove) are divisible by 5? 3'- /o'*A'Jz' JL,, Io--& =/m 01 q -? o1 arvr\ab t-q ' b , "* " Qq -- 9"t &or q 13. A computerpassword consists of 4 letters (A through Z) followed by a single digit (0 through9). Assume that the passwordsare casesensitive (i.e., uppercaseletters are considereddifferent from lowercase letters). passwords possible? - . a. how manydifferent are { J, 5,1 5 L' fa / o a )t qlpds ffi ----5\ b. how many different passwordsstart with Z? | er a-a -/ // "* I - {4&_* /p = L4oGPgo c. How many differant passwordsdo not start with eitherz or Z? 5o_ . , z.r2!,r-.p3lJ2r4dr---=-_- d. How many different passwordshave no Z's in them, either uppercaseor lowsrcase? to: IA oo-> -9**5o fo G el 2 ooD . %b- Ltq - P"{^.n J 14. A set of referencebooks consistsof 8 volumes numbered1 through 8. T"-r a. In how many ways can the 8 books be Erane6d on a shelf? 7of _8 +-3 f*' I =-8l- r' t2'?, b. In how many ways can the 8 booksbe aranged on the shelf so that at least I book is out of order? 4Dr3Jo-+ : (4orJrq.*- 15.A restaurantoffers a menu *rrT#rliioK'#ffidr#Tuifferent soups,4different satads, g different main counres,and 5_different desserts. x a. A fixed-price lunch meal consistsof a choice of appetizer,salad, qnd main course. How many differentlunch fixed-price meals are possible? =kaq--:-= -+=,+"j'^T' 1^,,.po,":' 5d. /\ctt\ J b. A fixed-price dinner meal consists of a choice of appetizer, a choice of soup or salad, a main course,g[ a dessert. How many different dinner fixed-price meals are possible? \-, - ! _ + vd.l"\,\fv C\R S"y or l^{n,'i lrrt _ :"a,\ n BoS c. The dinner special consistsof a choice'of sqgp oi salador both, -- and a mail cou$e. How many dinner):-- specials ^^j^l- ^-^are rl----^frtbere? -.q--ff--- 3 J,' t =:==::ffiF* -. v+)+-[p.+)J1 . =Eo,o)"1_ L(t 1 E.t _ +a4ot5^\,n tffi rvLq,lY\ 16.As part of a prime-timeTV garne,the player wins $1,000,000 if the first four cardsdrawn from a shuffled poker deck are an Ace, a King, a Quben, ffid a Jack (in any order). How many 4 card combinationsof this tlpe are "winning" foursomes? -'t'no a^r',gL '+e lk ti-[ t l ? - -i /frH Qy ori 17. If the player draws four.cards from the deck (samegame as above)and the cardsdrawn are an Ace, a King a Queen,and a Jack, in that order,the player wins the GRAND SLAM of $100,000,000.How many ways are thereto draw a GRAND SLAM in this game? +'4'+,4=W - F Kai-rJ 18. How many dilferent ways are there to draw 4 cardsfrom a poker deck, if the order they aredrawn in does not matter? t)) (5a-Cl+t..