Permuaions and Combinaions Combinaorics Copyrigh Sandards 006, Tes - ANSWERS Barry Mabillard. 0 www.mah0s.com
1. Deermine he middle erm in he expansion of ( a b) To ge he k-value for he middle erm, divide he exponen by : 10 =. Now apply he formula: 10- n-k k +1 = 10 C ( a) (-b) k+1 = n Ck ( a) ( b) = a -3b 6 6 =-806a b. In how many ways can 13 colleagues si a wo round conference ables if he firs able seas seven and he second able seas six? To begin, selec he seven colleagues ha will si a he firs able. This can be done in 13 C 7 =1716 ways. 7-1!=6!=70 ways. Then arrange hese seven people in a circle in Finally, ake he six remaining people, and arrange hem in a circle in ( 6-1 )!=!=10 ways. The answer is obained by muliplying hese resuls: 1716 70 10 = 18 6 00 3. Fred is having a pary and invies ou of his 8 friends. a) In how many ways can he choose which five people come o he pary? The order of friends aending he pary is no imporan. Use a combinaion. 8 C =6 b) In how many ways can he choose he five guess if Daniel and Sephanie mus eiher aend ogeher or no a all? If Daniel and Sephanie are ogeher, ha means ou of 6 friends (wo fewer remain in he group), Fred mus selec hree (since Daniel & Sephanie are already chosen) This can be done in 6 C 3 =0 ways. If Daniel and Sephanie are no coming o he pary, ha means here are 6 people available o choose from, and all posiions are sill available. This can be done in C =6 ways. 6 Add he cases o ge 6 ways. 10 Combinaorics Sandards Tes - ANSWERS 1 www.mah0s.com
. How many seven digi numbers are possible using 1,, 3,,, 6, and 7 if he middle digi is even and repeiions are no allowed? There are hree even digis which may go in he middle: Now fill in he remaining spaces wih he res of he digis. There are hree flighs from Edmonon o Winnipeg, and five flighs from Winnipeg o Torono. How many differen ways can a passenger fly from Edmonon o Torono hrough Winnipeg? Muliply he firs se of flighs (3) by he second se of flighs (). 3 = 1 6. How many numbers beween 100 and 1000 are odd? Repeiions are allowed. The las digi mus be 1, 3,, 7, or 9 for he number o be odd. Therefore, here are five possibiliies for his posiion. Now fill in he remaining posiions. There are 9 possibiliies for he firs digi (i can be zero), and 10 possibiliies for he second. 7. Deermine he sixh erm in he expansion of x 3 9 x n-k k n-k Use he formula = C a b. For he sixh erm, k = = C a b 9-3 +1 = 9 C ( x ) - x 3 6 =16( x ) - x 8 3 6 =16x - x 3 6 =-30618x k Combinaorics Sandards Tes - ANSWERS www.mah0s.com
8. In how many ways can leers from he word EDITOR be arranged if vowels and consonans alernae posiions? Firs deermine he number of arrangemens wih consonans firs: Then deermine he number of arrangemens wih vowels coming firs: Add he resuls o ge 7 possible arrangemens. 9. How many differen arrangemens of he word MATHEMATICS are here if he M s mus say ogeher? Pu a circle around he wo M s o indicae hey are one objec: Now evaluae he number of arrangemens: 10! =90700!! (We mus divide ou he repeaed A s and T s. We don need o divide ou he wo M s since hey have been combined ino one objec) 10. How many four digi numbers beween 999 and 9999 are divisible by and have no repeaed digis? There are wo separae cases ha mus be addressed. The firs case involves a five being he las digi, and he second case involves a zero being he las digi. Fill in he spaces as follows: Now fill in he remaining spaces: Add he resuls o ge he oal: 9 Combinaorics Sandards Tes - ANSWERS 3 www.mah0s.com
11. In he expansion of( x + y) 1, wha is he numerical coefficien of he erm conaining 3 11 x y n-k Use he formula = C a b. Firs predic wha k has o be in order o ge he required erm: +1 1 1- = C x y If he empy box is filled wih he number 11, you will ge he required erm. 11+1 1 11 1-11 11 = C x y 1 = 36x y 3 11 1. If here are six empy seas in a row a a movie, how many ways could four people fill hose seas? Arrangemen maers in his case, so use a permuaion. 6 P = 360 13. Solve for n: P = n 6 np=6 n! =6 (n- )! n(n-1)(n- )! =6 (n- )! n(n-1)=6 n -n=6 n -n-6=0 (n- 8)(n+7)= 0 n=8 Rejec -7 since you can have negaive objecs o selec from. 1. The head of a commiee needs o assign members o a sub-commiee. The posiions of presiden, vice-presiden, and reasurer mus be filled. In how many ways can Gary, John, Kevin, Ned, Sacy, and Veronica be assigned o hese hree posiions? Since iles are involved in he commiee, use a permuaion. There are 6 people available o fill hree posiions. 6 P 3 =10 k Combinaorics Sandards Tes - ANSWERS www.mah0s.com
1. Mark has songs and wishes o pu 1 of hem on a mix CD. a) In how many ways can he 1 songs be chosen? Order does no maer if he songs are merely being chosen. C = 700 1 b) The songs Corners and Predicions sound very similar. If Mark uses no more han one of hese songs on his mix CD, how many was can he choose he 1 songs for his CD? Wach ou for he wording no more han one. Tha means he could choose neiher song, or one song. The number of ways o selec he 1 songs wihou picking eiher Corners or Predicions is 3C 1 =817170. The number of ways o pick he 1 songs using one of he songs is C 1 3 C 13 = 8813 Add he resuls o ge he oal number of cases: 3103 16. Using he leers from he word PENCILS, a) Deermine he number of leer arrangemens 7P = 0 b) How many seven leer arrangemens are possible if N mus be he firs leer and he leers P & I mus be ogeher? Firs group he leers as follows: If N mus go firs, here is only one possibiliy for ha posiion. Ou of he objecs remaining (remember he bubble couns as only one objec), we can arrange hem as shown. However, don forge ha we can sill arrange wha s in he bubble in! = ways. The answer is 10 = 0 ways. Combinaorics Sandards Tes - ANSWERS www.mah0s.com
17. There are differen pahs from home o school, and 6 differen pahs from school o he convenience sore. How many ways can a suden go from home o he convenience sore, sopping a he school along he way? Muliply he number of pahs o he school () by he number of pahs o he convenience sore (6). 6 = 18. If here are hree differen cars and seven parking spaces, how many unique ways can he cars be parked? Use a permuaion since order is imporan: 7 P 3 =10 19. Solve for n algebraically: ( n+ 3)! = 0( n + 1)! (n+3)!= 0(n+1)! (n+3)(n+)(n+1)!= 0(n+1)! (n+3)(n+ )= 0 n +n+6=0 n +n-1=0 (n+7)(n- )= 0 n= Rejec -7 since n mus be a whole number 0. Given (3x + y) 8, deermine he middle erm of he expansion. The k for he middle erm is 8 = 8C (3x) (y) =70(3x) (y) 8- =70(81x )(16y ) =9070x y 1. A eacher wishes o have a picure aken wih six eam players. In how many ways can hey line up for he picure if he eacher mus sand in he middle? Combinaorics Sandards Tes - ANSWERS 6 www.mah0s.com
. How many four digi posiive numbers less han 670 can be formed using he digis 1, 3,,, 8, 9 if repeiion is no allowed. We need o separae his quesion ou ino differen cases since numbers in he 000 s have exra resricions. Case 1: Numbers in he 000 s: There is only one possibiliy for he firs digi {}. The nex digi has hree possibiliies. {1, 3, }. There are possibiliies for he nex digi since any remaining number can be used, and 3 possibiliies for he las digi. Case : Numbers in he 1000 s and 3000 s: There are wo possibiliies for he firs digi {1, 3}. Anyhing goes for he remaining digis, so here are, hen, hen 3 possibiliies. Add he resuls ogeher: 36 + 10 = 16 3. In how many ways can he leers in he word QUILT be arranged if he vowels and consonans alernae? Since here are hree consonans and wo vowels, here is only one opion. I mus go CVCVC. Simplify: C C 00 300 00 100 C C 00 300 00 100 = 00! 00-300!300! 00! ( 00-100 )!100! 00! ( 00-300 )!300! 00! = 00! 300! 100! = 100! 300! 00! =1 00-100!100! Combinaorics Sandards Tes - ANSWERS 7 www.mah0s.com
. Using all he leers from he word SELECTIONS, how many disinguishable arrangemens can be made? There are 10 leers. There are E s repeaed and wo S s repeaed. 10! 10! The number of arrangemens is = = 90700!!!! 6. Given (8x 6 7y 3 ) 9, deermine he posiion of he erm conaining x 36 Firs find he k value of he erm conaining x 36 6 9-3 9C (8x ) (-7y ) By inspecion, we'll ge x when k = 3 36 This corresponds o he fourh erm of he expansion since he k- value is always one less han he erm posiion. 7. Cassie, Richard, and 7 of heir friends aend a movie. In how many ways can hey be seaed so ha Cassie and Richard do no si nex o each oher? To find he number of ways Cassie and Richard can sand nex o each oher in line, rea Cassie & Richard as a single iem. The number of arrangemens is 8!! = 8060. (Don forge o use he! since Cassie & Richard can be arranged wo differen ways inside heir bubble.) The number of ways nine people can be arranged wihou resricions is 9! = 36880 The number of ways Cassie and Richard do no sand nex o each oher in line is 36880 8060 = 80 8. In how many ways can six people be seaed a a round able? Use he formula for a circle permuaion ( n-1 )! The answer is ( 6-1 )!=!=10 Combinaorics Sandards Tes - ANSWERS 8 www.mah0s.com
9. The fifh erm in he expansion of a 3 n a conains a. Deermine he value of n. n-k Use he formula = C a b. n- 3 = C a - a To ge he fifh erm, k =. +1 n By inspecion, we can see ha n = 6 yields he correc exponen for a. 6-3 = 6 C( a ) - a 3 =1( a ) - a 8 81 =1a - a =-11a 30. Solve for n in he equaion ( n + )! = 8 ( n + 1)! k Exra seps o show he correc exponen is obained. (n+)! =8 (n+1)! (n+)(n+1)! =8 (n+1)! n+=8 n=6 31. How many erms are in he expansion of x + x 3 1 7 The number of erms in an expansion is one more han he value of n. Therefore, here are 8 erms. 3. In how many ways can six sudens si in a row if wo sudens, Jerry and Jennifer, mus si ogeher? We can arrange he row in!!= 0 ways. Combinaorics Sandards Tes - ANSWERS 9 www.mah0s.com
33. Find and simplify he fourh erm in he expansion of ( a 3b) 6 n-k Use he formula = C a b. To ge he fourh erm, k = 3. 3+1 6 3 6-3 3 = C a -3b 3 3 =0 a -3b 3 3 = 0 8a -7b =-30a b 3 3 3. How many four leer arrangemens can be made from he leers DDEFGH This quesion is ricky since we have o look a hree cases. The case where no D is chosen: Here we wish o arrange he leers EFGH, which can be done in! = ways. The case where one D is chosen: If a D is seleced, hen we need o choose hree addiional leers from EFGH. This can be done in C 3 ways. Now ha we have seleced all four leers, arrange hem in! ways. The number of cases where one D is seleced is C!=96 3 The case where wo D s are chosen: If wo D s are seleced, hen we sill need o selec wo leers from EFGH. This can be done in C ways. Now ha we have chosen all four leers, arrange hem in! ways. Don forge ha since we have wo D s, we need o divide ou repeiions!! The number of cases where wo D s are seleced is C =7.! Add up all he separae cases: + 96 + 7 = 19 k *If you hough you can use 6P, his is incorrec because i only works if repeaed leers are presen in! all arrangemens. Obviously an arrangemen like DFGH would no need he division! Combinaorics Sandards Tes - ANSWERS 10 www.mah0s.com
3 3. The erm 1080a b occurs in he expansion of ( a 3b) n The value of n is: n-k Use he formula = C a b. Firs predic wha k has o be in order o ge he required erm: To ge back b 3, here is only one opion for k. I has o be 3. 3+1 n 3 n-3 3 = C a b To ge back a, he value of n mus be 3+1 3 3-3 3 = C a -3b -3 3 = C a -3b 3 =10 a -7b =-1080a b 3 k 36. There are 1 musicians o be separaed ino hree groups of five. The players for Group 1 are chosen firs, hen Group and finally he Group 3. How many ways can his be done? The players for Group 1 can be chosen in 1 C = 3003 ways. The players for Group can be chosen in 10C = ways, since only 10 people remain o be seleced. The players from Group 3 can be chosen in C =1 way. The oal number of ways is 3003 1 = 7676 ways. 37. A grad commiee of sudens is o be formed from a class of 1 sudens. Danielle, Francis, and Hillary decide hey do no wan o be on he commiee. How many commiees can be formed? (Express answer as facorials) If hese hree people are no going o be on he commiee, hey can be removed from he selecion pool. As a resul, here are 18 sudens sill in he selecion pool, and all posiions mus be filled. 18! 18! This can be done in 18C = = 18 -!! 13!! ways. Combinaorics Sandards Tes - ANSWERS 11 www.mah0s.com
a ) 11 38. The expression ( is expanded using descending powers of a. Deermine he posiion of he erm conaining a n-k Use he formula = C a b. Se i up wih he known informaion: = C ( a) (-) k k+1 11 k 11-k k To ge back a, here is only one opion for k. I has o be 9. 9+1 11 9 10 = a -1 10 11-9 9 = C a - =-8160a The erm conaining a is he enh erm. 39. If P = 670 and C = 6, hen he value of r is n r n r Wrie n Wrie P r =670 as n C = 6 r as n! =670 n-r! n! =6 n-r!r! Now divide he expressions o simplify n! ( n-r )! n! ( n-r )!r! 670 = = r! On he righ side, =10 n! ( n-r )! n! 6 n-r!r! r! = 10 when r =. The answer is. Combinaorics Sandards Tes - ANSWERS 1 www.mah0s.com