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# Intermediate Algebra: Ch. 10.1 10.3

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• 1. HHl"y,9,v,,*c"#i*. fm Use the fundamental eounting principle and permutations. Vocabulary An ordering ofz objects a permutationofthe objects. is The symbol ! is the factorial slrrnboldefinedasfollows: nr = 2. (n - l). (, - 2)..... 3 . 2 . lf\$ffinm Use the fundamental counting principle Dining.A restaurant offers5 appetizers,4 salads, entrees, 7 desserts. 11 and How manydifferentwayscanyou ordera completemeal? Solution Youcaausethe firndamentalcountingprinciple to find the total numberofways to ordera completemeal.Multiply the numberof appetizers the number (5), of salads (4), thenumber entrees of (11),andthenumber oia".r"rts 17;. Nu mber ofways:5 . 4. 11.7 :1 5 4 0 Thenumber different of waysyou canordera complete mealis 1540.ffirffi Use the counting principle with repetition Fundraising A raffle ticket contains1 digit followed by 3 letters. (a) How many arepossible letters digits berepeateai nowmany if and can different :lFl arepossibleif lettersand digits carurotU" ."peut")-i 9lP::11tickets 1u; Solution a. Therere 26 choices each for letterand 10choices eachdigit. for . You canusethe fundamental countingprinciple to find the number of differenttickets. Number oftickets: l0 . 26 . 26 .26 : 175,760 With repetition, number different the of ticketsis 175,760. b. If you cannotrepeatletters,thereare26 choicesfor the first letter, but then only 25 choicesfor the secorld letter,and24 choicesfbr the third letter Therearestill l0 choicesfor the digit. you canuse the fundamental countingprinciple to find the nuniberof different tickets. Number oftickets: l0 . 26 . ZS. 24: 156,000 Withoutrepetition, number different the of ticketsis 156,000. Exercises for Examples 1 and 2 1. A deli sells4 sizes ice cream of cones (small,medium, large, giant)and and 2 different cones(waffle and cake).How manychoicesdoJsthe Oetioffert 2, { 1aIfleticket conlains4 digits followed by 2 leters. (a) How manydifferent tickets arepossibleif lettersand digits canbe repeatedi(b) How manydifferent ticketsarepossible letters digitscannot repeatedi if and be Algebra 2 . Chapter Hesource 10 Book
• 2. Name@f,,.,y#t,9",H-,*" Find the likelihood that an event will occur. Vocabulary. The probability of an eventis a numberfrom 0 to 1 that indicatesthe likelihood event the will occur. When all outcomes equallylikely, the theoretical probability that are an eventwill occuris Numberofoutcomesin event I r(4 ) : --Totuftffier of ouGomes Odds measure likelihood that an eventwill occur. the The experimental probability of an eventI is givenby : Numberoftrials where occurs I f(li - Totul nGbet o-fttiul, A geometricprobability is a ratio of two lengths,areas, volumes. or m@ Find probabilities of events There ar€ eight balls in a bag. Three are red, three are yellow, and two a?6green. Find the probability of choosing a gteen ball. Thereare 8 possibleoutcomes. Number of ways to choose a green ball 2l I(choosinga greenball) : Numberof outcomes 84 Etrtr[E&:lUse permutationsor combinations Auditions For a play tryout, 10 students reciting monologuesThe orderin which are the studentsperfom is randomlyselected. What is the probability that the students (a) auditionin alphabetical orderby last name?(b) What is the probability that 2 of your 4 friendstrying otlt will be the fust 2 performers? a, Thereare10! differentp of Only l is ermutations the 10 studenls. orderby last name. in alphabetical ll order): l-or : t6r&800 - 0.000000276 P(alphabetical , b. Thereare ,oC, different combinationsoftwo students. these Of oC, are2 ofyour friends. ,C" A ) P(fust2 students your friendsl: = are Fr:; 15- 0.133 Exercises for Examples 1 and 2 1. Find the probability of choosinggnered ball in Example I . 2. Find the probability that the fust 3 performersareyour ftiends in Example2. 2 Algabra Chapter Resource 10 Book 3t
• 3. Name Date Guida r -rl,,l ,studypages682-\$89 "o,t,,,o L_j!:!_l Forusewith mflntr Findthe numberof permutations Track Meet Eight runnerscompete an elite track meetcompetition.(a) Assuming in thereareio ties, in how manywayscould the runnersfinish in the meet?(b) Assuming thereareno ties, in how manywayscould the runnersplacefirst, secon4or third in themeet? Solution a. Thereare8! differentwaysthat the runnerscanfinish. 8! : 8. 7 . 6. s. 4 . 3 . 2 . 1 : 4 0 , 3 2 0 b. Any of the 8 rumers canfinish fust, then any of the remaining 7 runnerscanfinish second, any ofthe remaining6 runners and canfinish third. So the numberof waysthat the runnerscanplace second thirdis 8.7 .6:336. first. or ruEru Find permutations of .n obiects taken r at a time Baseball A baseballteamhas 14players.Hbw many9-playerbatting orderscan be formed? Solution Find the numberofpermutationsof 14playerstaken9 at a time. tnn l4! = ]4 - 87,178.,ot : P - 726,485760 e r 14 - 9)! 14: st t;::1 The team can form 726,485,760 batting orders with 9 players. HU!ffi Find permutataonswith repetition Find the number of distinguishable permutations of the letters in (al DELAWAREand (bl PHILADELPHIA. a. DELAW.ARE 8 lettersofwhich A and E areeachrepeated has 2 times.So,thenumber distinguishable of permutations is R 40.320 2,.at= q : 10080 b. PHILADELPHIAhas12letters whichB H, I, L, andA are of eachrepeated times. So,the numberof distinguishable 2 . t2! 479qo 600: 14.968,800. ls Permurallons - t! .2r. . 2r. 2r . 2! Exercises for Examples 3,4, and 5 3. ReworkExample3 if 9 runnerscompete. 4. ReworkExample if thebaseball 4 teamhasl2 players. Find the number of distinguishable permutations of the !€tters in the word. 5. ALABAMA 6. STREET 7. MISSOURI Algebra 2t0 ChapterResource 10 Book
• 4. Name .n.l-Study GuidG) continued f | .v.- | Foruse with pages 690_\$97 Exercises for Examples 1 and 2 Find the number of combinations. 1, uCo 2. ,oCt 3. oC" 4 ,tcr. 5. ReworkExample2 to find how manywaysyou cango to at leasttwo of the four football eames. Use Pascals triangle CommitteeMembers Usepascalttriangle find thenumber combinations to of of 3 committeememberschosenfrom g availablemembers. Solution To find rC, write the 8th row of pascal,s triangle. n:7 (Tthrow) 1 721 353521 71 z:8 (8throw) I 8 28 s67056288 I rco ,c, ,c, ,c, ,Co ,c, ,cu ,c, ,C, ThevalueofrC, is the4th number the gthrow ofpascal,s in triangle, so = 56. ,C, Thereare 56 combinations 3 committeemembers. of mGElfi Expanda power of a binomial difference Use the binomial theorem to write the binomial expansion. tz - ?t l = I u + r - 1113 J_,I t- : 3c0t(-3)0 + 3ct*{ rt + 3c2zt(-3)2 3q{.r3 + = (rxrxi) + (r(?)C, + (3XzXe) + (1XlX_27) :23-9?+272- 2 7 trWrffii Find a coefficient in an expansion Find the coefficient of x3 in the expanaion of (4x + 315. Eachtermin theexpansion the form has - ,e)r.The termcontaining:r3 sc!4is occurswhenr:2: = sc2(4i3(3)2 0o)(64x3)( = 576ox3 Thecoefficient -r3is 5760. of Exercisesfor Examples 3,4, and S 6. ReworkExample3 choosing4 committeemembers. 7. Usethebinomialtheorem expand expression + Z)a. to the (x3 8. Findthe coeffrcient the.x2_term Example . of in 5 Algebra 22 ChapterResource 10 Book
• 5. Name , -di.l ,studypages6 | .v.! | Forusewith Guid€ "o,r,,"a -704 milqr+ Find odds A six-sided dieis tossed. Find (al the odds in favor of getting a 5 and (bl the odds agarnst getting a 5. a. Odds in favor ofgetting a 5 : Number of fives | .- Number ofnon-fives i, or r:) b. Odds against gelling 5 a : f. or 5:t Ifn:fililtfi! Find an experimental probability The table shows the results of tossing two coins twenty times. Find the experimental probability of getting a head on the first toss. HH 3 Theexperimental p probability ofgettinga headon HT 6 thefust tossis the sumof HH andHT. ?+ 5 0 TH 5 fl nead on lst toss) -- - -20" = : 0.45 20 TT 6 ,ly.Fr,nf*t Find a geometric probability You throw a beanbag at a square board shown. F_4ft __ 1 Your beanbag is equally likely to hit any point on the board. ls the bag more likely to land outside T the smaller square, or inside the smaller square? Areaoutside smailsquare JN P(landing outsidesmall square)- Areaofentireboard _42 -32 _I I I 42 16 P(landing inside small = A:i++-:d]j!"= square) : { : : Areaol entireboard 42 l6 because < -7 9 t6 16.you aremorelikely to landinsidethesmaller square. Exercisesfor Examples 3,4, and 5 A card is drawn from a standard deck. Find the odds. 3. Againstdrawing king a 4. In favorof drawing club a 5. Findtheexperimental probability gettinga tail on the second in of toss Example4. 6, ReworkExample for squares 5 with sides 2 feetand3 feet. of Algebra232 Chapter Besource 10 Book