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# Class Note for MATH 1313 with Professor Flagg at UH

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This 9 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 19 views.

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Date Created: 02/06/15

Math 1313 Section 64 Permutations and Combinations In this section we will consider the number of ways in which the elements in a set can be arranged Before we can determine the number of possible arrangements we have to know whether or not order matters In other words if our set is 1 2 3 and we are looking for all possible arrangements of the elements of the set and order matters then 123 and 231 are different arrangements However if order does not matter then 123 and 231 would be considered to be the same When does order matter Suppose you and five of your friends are lining up to take a group photo How many ways can this happen In this case order matters You ll get a different photo if you are first in line than you will if you are last in line Or suppose you are in line with five other people In this case order matters Now suppose you are checking your Cash 5 lottery ticket and you see that you have the correct five numbers You are still a winner even if the numbers on your ticket are in a different order than the order in which they were drawn In this game order does not matter The group photo example is an example of a permutation order matters and the lottery example is an example of a combination order does not matter De nition n is read 11 factorial and is defined to be nniln72n7332l Then6654321 7amp0 Math 1313 Class Notes 7 Section 64 Page 1 of9 ag NPK A Ca 27 W You can work this out on your calculator with just a couple of key strokes Be sure you nd the factorial key on your calculator and that you know how to use it Permutations We will consider several types of permutation problems First we ll look at permutations of all of the elements of the set Example 1 How many possible photos are there if siX people line up in single le to take a picture Q39i39 39 i39m ref 7510 l The number of permutations of n distinct objects taken n at a time written Pn n is n You may also see this notation for permutations quotPnwhich also means nl NW 011 07Ck ExampleZ FindP8 8 form h picc raw 8 l 40 3 a O P1 Uf Z While you can compute this using factorials it is sometimes more convenient to use the permutation feature on your calculator You need to nd this key and be sure you know how to compute permutations using your calculator Math 1313 Class Notes 7 Section 64 Page 2 of9 Often we want to consider arrangements of less than all of the elements from a set For example suppose there are 27 people in a club and the club is electing new officers They will elect a B epresident Vicepresident secretary and treasurer We want to o know the number of possible ways that this can happen Order matters since being elected president is not the same as being elected treasurer In this case the number of permutations of n objects taken r at n n 7 r I You may also see this notation for permutations P which also anr means fty N Pr P 5N Q 39Q391 3 mark 3 Example3 FindP10 3 0 ww lt7 9 l0 0 orgW 1 MM 0 3 7 r C I g 397 a 0 You can also work problems like these on your calculator a time where r g n is given by Pn r Example 4 A club with 27 members will elect new officers They will chose a president Vicepresident secretary and treasurer In how many ways can this happen s 20 IL Ll f oli ized 06 9ch 0me 0119f P74 49QOO Math 1313 Class Notes 7 Section 64 Page 3 of9 Example 5 Eight people apply for three different jobs Each applicant is quali ed to ll each of the three jobs If the jobs are lled from these 8 people in how many ways can the jobs be lled N g r 3 Mger jd s 5 parwo iojlilon P a3j 334 The nal type of permutation problem involves a set where the objects in the set are not all distinct objects ie there are repeated or identical items in the set Given a set of n objects not all distinct with nl objects alike nZ objects alike n39 objects alike and n1 nZ n n then the number of permutations of these It objects taken n at a time is O i 2 1 z r39 Example 6 How many permutations of the letters in the word PEPPER are there Q I 5772ltPE 63 3 F F a R w 41ga ivw G 39 1 quot I P 0 Example 7 The Fall Flagg Co would like to create a company password by arranging the letters in the company name FALLFLAGG How many arrangements are possible Fe P 7 Aquot QC al QI Q t jggaiiQ t mseo G Q Q39R Math 1313 Cla 5 Notes 7 Section 64 Page 4 of9 3 1 g Combinations We will use combinations throughout the rest of the course You should become very pro cient at working with combinations on your calculator The number of combinations of n distinct objects taken r at a time r gn is given by Cn r quotI rn 7 r I You may also see this notation for combinations C which also 5nr5 m wquot N Cg QLM5 in Ao l 29 CASH ca LE CO QKO of J Camel mo 4 6quot O l 4 Example 9 How many 5card hands can be dealt from a standard deck of 52 playing cards CsaI s 1576 7Q 0 Example 8 Find C104 Example 10 A sixmember governing council will be chosen from a 40member group How many possible committees are h 7 tere COTVLMz39HeeE cam lamalvon C40 03 t 39333 3 W9 Math 1313 Class Notes 7 Section 64 Page 5 of9 Sometimes problems will involve the use of the multiplication principle In these problems we have more than one task We ll use permutations or combinations to determine the number of ways in which each task can be completed then multiply these answers together Example 11 A company car has a seating capacity of 6 and will be used by a 6person carpool If only 4 of the employees will drive how many possible arrangements are there Per m Esk l CLGQS17 DPv r T j v 4 f auxm7 FaSSW J rDoNae lt F PasseW 4 x 3 b 5 Example 12 A committee is made up of 10 men and 8 women A subcommittee will be formed and it must have men and 3 men as members In how many ways can this happen Conb MQLQdq H 0rampe 7 k t i iiiikrbwm Pigf Mm CF61 6003 367 quot 9 0 amp7amp0 Math 1313 Class Notes 7 Section 64 Page 6 of9 Example 13 A student belongs to a media club This month he must purchase 3 CDs and 4 DVDs from the club s featured list There are 12 featured CDs and 12 featured DVDs In how many ways can he select his purchase comb IiIa enq ask I 73 2 CD39 DVD 5 CQ Q 10242 310 v L6 95 Note that these questions ask how many ways we can do two or more specific tasks in succession like picking 3 CD s out of 12 or 3 men out of 10 to pick from Sometimes the questions we want to ask involve more than one specific task I MAAFDL 7 lifen5 3CDs m fCDS 3CD 1 LE CD 5 3m DVD 71 4CD 4d 3m CQ 3l Cla f 1L C0214 C 2157 Example 14 A coin is tossed 5 times In how many ways can at least 4 heads appear h 4 6W 6 5 W HSHHT HHHHH HHTH CA g l H HEW quotr THHHHC5LQ T0421 Math 1313 Class Notes 7 section 64 Page 7 of9 3 I 2 Q ochqu glammm Atmostmeansi a M Q 3 0391 QoJ 0 I 21 Example 15 A hardware store receives a shipment of 100 battery packs A A sample of 4 batteries is taken for testing How many ways can a sample of 4 batteries be chosen COM m c too L0 37621 816 B Suppose 9 of the battery packs are defective How many ways can the selection of 4 batteries for testing contain 2 Q defective battery packs H0 5L Bank a GoctQ amp 7v c 7 a O Caws acquot QMg 3 1 C How many selections contain at most one defective battery T M0571 DEF 3 e446 05 a 225 0042 234 3 6 00 Mano Cq l C 71 n C 243 I amamo 2 i39 QlLf 7c57035 D How many selections o 39 3 At least means 7 in at least one defective A70 bagwtm a 3 m 4f Mm zswo oz 37 31on 076 3 M5 emu ore 4 mg 6va 0 60quot x4 GoaQ 39 74w tweeaMPcket giggi fogo ii m gt W 339FQQS COW40 C 0 0 9786 O Math 1313 Class Notes 7 Section 64 Page 8 of9 7 1ampL 39555 Example 16 cards are drawn from a wellshuf ed deck of 35 52 playing cards How many ways can the draw contain at least 9 ad a red cards many s 4mg Laek 0L 5 HQMIMM 0amp 659 09ch cgcf ac ea 4 dams c 526 c9 g4JrM 9 5 18039amp 23q 3 39 SO 3 tr Qt ago DieltC QFWM394 MOT COQQB quot C 0 a M C13O 005391460 Example 17 A bag contains 13 balls 7 4 red 5 black and 4 green Five balls are chosen at random How many selections contain at least 3 b1 ck balls M9 3Elaxzi gm OK L fLae 39 1M 0 SM C6 l3gt C D CSLF d2 MUS5 fO Q8 4 58 3521 I I I Math 1313 Class Notes 7 Section 64 Page 9 of9

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