THE PRODUCT PRINCIPLE

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1 214 OUNTING ND THE INOMIL EXNSION (hapter 8) OENING ROLEM t an I Mathematics Teachers onference there are 273 delegates present. The organising committee consists of 10 people. ² If each committee memer shakes hands with every other committee memer, how many handshakes take place? an a 10-sided convex polygon e used to solve this prolem? ² If all 273 delegates shake hands with all other delegates, how many handshakes take place now? The Opening rolem is an example of a counting prolem. The following exercises will help us to solve counting prolems without having to list and count the possiilities one y one. To do this we will examine: ² the product principle ² counting permutations ² counting cominations THE RODUT RINILE Suppose that there are three towns, and and that 4 different roads could e taken from to and two different roads from to. We can show this in a diagram: How many different pathways are there from to going through? If we take road 1, there are two alternative roads to complete our trip. If we take road 2, there are two alternative roads to complete our trip,... and so on. So, there are =4 2 different pathways. Notice that the 4 corresponds to the numer of roads from to and the 2 corresponds to the numer of roads from to. road 1 road 2 D Similarly, for: there would e 4 2 3=24 different pathways from to D passing through and. THE RODUT RINILE If there are m different ways of performing an operation and for each of these there are n different ways of performing a second independent operation, then there are mn different ways of performing the two operations in succession. road 1 road 2

2 OUNTING ND THE INOMIL EXNSION (hapter 8) 215 The product principle can e extended to three or more successive operations. Example 1 R S It is possile to take five different paths from auline s to uinton s, 4 different paths from uinton s to Reiko s and 3 different paths from Reiko s to Sam s. How many different pathways could e taken from auline s to Sam s via uinton s and Reiko s? The total numer of different pathways = = 60. fproduct principleg EXERISE 8 1 The illustration shows the possile map routes for a us service which goes from to S through oth R and R. How many different routes are possile? S 2 It is decided to lael the vertices of a rectangle with the letters,, and D. In how many ways is this possile if: a they are to e in clockwise alphaetical order they are to e in alphaetical order c they are to e in random order? 3 The figure alongside is ox-shaped and made of wire. n ant crawls along the wire from to. How many different paths of shortest length lead from to? 4 In how many different ways can the top two positions e filled in a tale tennis competition of 7 teams? 5 footall competition is organised etween 8 teams. In how many ways is it possile to fill the top 4 places in order of premiership points otained? 6 How many 3-digit numers can e formed using the digits 2, 3, 4, 5 and 6: a as often as desired once only? 7 How many different alpha-numeric plates for motor car registration can e made if the first 3 places are English alphaet letters and those remaining are 3 digits from 0 to 9? 8 In how many ways can: a 2 letters e mailed into 2 mail oxes 2 letters e mailed into 3 mail oxes c 4 letters e mailed into 3 mail oxes?

3 216 OUNTING ND THE INOMIL EXNSION (hapter 8) OUNTING THS onsider the following road system leading from to: D From to there are 2 paths. From to there are 3 2=6 paths. From to there are 3 paths. Thus, from to there are 2+6+3=11 paths. E F G Notice that: I when going from to G, we go from to E and then from E to G, and we multiply the possiilities, I when going from to, we must first go from to, or to or to, and we add the possiilities. onsequently: ² the word and suggests multiplying the possiilities ² the word or suggests adding the possiilities. Example 2 How many different paths lead from to? E F D G H I Going from to to to to there are 2 3=6 paths or from to D to E to F to there are 2 paths or from to D to G to H to I to there are 2 2=4 paths. So, we have 6+2+4=12 different paths. EXERISE 8 1 How many different paths lead from to? a c d

4 OUNTING ND THE INOMIL EXNSION (hapter 8) 217 FTORIL NOTTION In prolems involving counting, products of consecutive positive integers are common. For example, or FTORIL NOTTION For convenience, we introduce factorial numers to represent the products of consecutive positive integers. For example, the product can e written as 6!. In general, for n > 1, n! is the product of the first n positive integers. n! =n(n 1)(n 2)(n 3):::: n! is read n factorial. Notice that can e written using factorial numers only as 8 7 6= ROERTIES OF FTORIL NUMERS The factorial rule is n! =n (n 1)! for n > 1 which can e extended to n! =n(n 1)(n 2)! and so on. Using the factorial rule with n =1, we have 1! = 1 0! We hence define 0! = 1 = 8! 5! Example 3 What integer is equal to: a 4! 5! 3! c 7! 4! 3!? a 4! = =24 5! 3! = =5 4= c 7! 4! 3! = =35 Example 4 Express in factorial form: a a = = 10! 6! = = 10! 4! 6!

5 218 OUNTING ND THE INOMIL EXNSION (hapter 8) Example 5 Write the following sums and differences as a product y factorising: a 8! + 6! 10! 9! + 8! a 8! + 6! 10! 9! + 8! =8 7 6! + 6! = ! 9 8! + 8! = 6!( ) = 8!(90 9+1) =6! 57 = 8! 82 Example 6 Simplify 7! 6! 6 using factorisation. 7! 6! 6 7 6! 6! = 6 6!(7 1) = 6 =6! 1 1 EXERISE 8 1 Find n! for n =0, 1, 2, 3, ::::, Simplify without using a calculator: 6! 6! 6! a c 5! 4! 7! d 4! 6! e 100! 99! f 7! 5! 2! 3 Simplify: a n! (n 1)! (n + 2)! n! c (n + 1)! (n 1)! 4 Express in factorial form: a c d e f Write as a product using factorisation: a 5! + 4! 11! 10! c 6! + 8! d 12! 10! e 9! + 8! + 7! f 7! 6! + 8! g 12! 2 11! h 3 9! + 5 8! 6 Simplify using factorisation: a e 12! 11! 11 6! + 5! 4! 4! f 10! + 9! 11 n!+(n 1)! (n 1)! c g 10! 8! 89 n! (n 1)! n 1 d h 10! 9! 9! (n + 2)! + (n + 1)! n +3

6 OUNTING ND THE INOMIL EXNSION (hapter 8) 219 D ERMUTTIONS permutation of a group of symols is any arrangement of those symols in a definite order. For example, is a permutation on the symols, and in which all three of them are used. We say the symols are taken 3 at a time. Notice that,,,,, are all the different permutations on the symols, and taken 3 at a time. In this exercise we are concerned with the listing of all permutations, and then learning to count how many permutations there are without having to list them all. Example 7 List all the permutations on the symols, and R when they are taken: a 1 at a time 2 at a time c 3 at a time. a,, R R R R R c R R R R R R Example 8 List all permutations on the symols W, X, Y and Z taken 4 at a time. WXYZ WXZY WYXZ WYZX WZXY WZYX XWYZ XWZY XYWZ XYZW XZYW XZWY YWXZ YWZX YXWZ YXZW YZWX YZXW ZWXY ZWYX ZXWY ZXYW ZYWX ZYXW i.e., 24 of them. For large numers of symols listing the complete set of permutations is asurd. However, we can still count them in the following way. In Example 8 there were 4 positions to fill: In the 1st position, any of the 4 symols could e used. This leaves any of 3 symols to go in the 2nd position, which leaves any of 2 symols to go in the 3rd position. The remaining symol must go in the 4th position. So, the total numer of permutations = =24 1st 2nd 3rd 4th 4 1st 2nd 3rd 4th st 2nd 3rd 4th st 2nd 3rd 4th fproduct principleg

7 220 OUNTING ND THE INOMIL EXNSION (hapter 8) Example 9 If a chess association has 16 teams, in how many different ways could the top 8 positions e filled on the competition ladder? ny of the 16 teams could fill the top position. ny of the remaining 15 teams could fill the 2nd position. ny of the remaining 14 teams could fill the 3rd position.. ny of the remaining 9 teams could fill the 8th position. i.e., st 2nd 3rd 4th 5th 6th 7th 8th ) total numer = = Example 10 Suppose you have the alphaet locks,,, D and E and they are placed in a row. For example you could have: DE a How many different permutations could you have? How many permutations end in? c How many permutations have the form ? d How many egin and end with a vowel, i.e., or E? a There are 5 letters taken 5 at a time. ) total numer = = any others here here must e in the last position (1 way) and the other 4 letters could go into the remaining 4 places in 4! ways. ) total numer =1 4! = 24 ways. c goes into 1 place, goes into 1 place and the remaining 3 letters go into the remaining 3 places in 3! ways. ) total numer =1 1 3! = 6 ways. d ore remainder of or E or E could go into the 1st position, and after that one is placed, the other one goes into the last position. The remaining 3 letters could e arranged in 3! ways in the 3 remaining positions. ) total numer =2 1 3! = 12:

8 OUNTING ND THE INOMIL EXNSION (hapter 8) 221 Example 11 There are 6 different ooks arranged in a row on a shelf. In how many ways can two of the ooks, and e together? Method 1: We could have any of the following locations for and 9 If we consider any one of these, the remaining >= 4 ooks could e placed 10 of these in 4! different orderings ) total numer of ways =10 4! = 240: >; Method 2: and can e put together in 2! ways (i.e., or ). Now consider this pairing as one ook (effectively tying a string around them) which together with the other 4 ooks can e ordered in 5! different ways. ) total numer =2! 5! = 240: EXERISE 8D 1 List the set of all permutations on the symols W, X, Y and Z taken: a 1 at a time two at a time c three at a time. Note: Example 8 has them taken 4 at a time. 2 List the set of all permutations on the symols,,, D and E taken: a 2 at a time 3 at a time. 3 In how many ways can: a 5 different ooks e arranged on a shelf 3 different paintings, from a collection of 8, e chosen and hung in a row c a signal consisting of 4 coloured flags e made if there are 10 different flags to choose from? 4 Suppose you have 4 different coloured flags. How many different signals could you make using: a 2 flags only 3 flags only c 2 or 3 flags? 5 How many different permutations of the letters,,, D, E and F are there if each letter can e used once only? How many of these: a end in ED egin with F and end with c egin and end with a vowel (i.e., or E)? 6 How many 3-digit numers can e constructed from the digits 1, 2, 3, 4, 5, 6 and 7 if each digit may e used: a as often as desired only once c once only and the numer is odd?

9 222 OUNTING ND THE INOMIL EXNSION (hapter 8) 7 In how many ways can 3 oys and 3 girls e arranged in a row of 6 seats? In how many of these ways do the oys and girls alternate? 8 3-digit numers are constructed from the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 using each digit at most once. How many such numers: a can e constructed end in 5 c end in 0 d are divisile y 5? 9 In how many ways can 5 different ooks e arranged on a shelf if: a there are no restrictions ooks X and Y must e together c ooks X and Y are never together? 10 group of 10 students sit randomly in a row of 10 chairs. In how many ways can this e done if: a there are no restrictions 3 students, and are always seated together? 11 How many three-digit numers, in which no two digits are the same, can e made using the digits 0, 1, 3, 5, 8 if: a there are no restrictions the numers must e less than 500 c the numers must e even and greater than 300? 12 How many different arrangements of four letters chosen from the letters of the word MONDY are possile if: a there are no restrictions at least one vowel ( or O) must e used c no two vowels are adjacent? 13 Nine oxes are each laelled with a different whole numer from 1 to 9. Five people are allowed to take one ox each. In how many different ways can this e done if: a there are no restrictions the first three people decide that they will take even numered oxes? 14 lice has ooked ten adjacent front-row seats for a asketall game for herself and nine friends. ltogether, there are five oys and five girls. a ssuming they all arrive, how many different arrangements are there if: i there are no restrictions ii oys and girls are to sit alternately? Due to a severe snowstorm, only five of lice s friends are ale to join her for the game. How many different ways are there of seating in the 10 seats if: i there are no restrictions ii any two of lice s friends are to sit next to her? 15 t a restaurant, a rectangular tale seats eight people, four on each of the longer sides. Eight diners sit at the tale. How many different seating arrangements are there if: a there are no restrictions two particular people wish to sit directly opposite each other c two particular people wish to sit on the same side of the tale, next to each other?

10 OUNTING ND THE INOMIL EXNSION (hapter 8) 223 INVESTIGTION 1 Notice that ERMUTTIONS IN IRLE There are 6 permutations on the symols, and in a line. These are:. However in a circle there are only 2 different permutations on these 3 symols. These are: What to do: and are the same cyclic permutations. 1 Draw diagrams showing different cyclic permutations for: a one symol: two symols: and c three symols:, and d four symols:,, and D 2 opy and complete: Numer of symols ermutations in a line ermutations in a circle =3! 2=2! 4 3 If there are n symols to e permuted in a circle, how many different orderings are possile? as they are the only possiilities with different right-hand and lefthand neighours. E OMINTIONS comination is a selection of ojects without regard to order or arrangement. For example, the possile teams of 3 people selected from,,, D and E are D E D E DE D E DE DE i.e., 10 different cominations. n r is the numer of cominations on n distinct or different symols taken r at a time. n r the numer up for selection the numer of positions needed to e filled. n r may also e written as n r or as the inomial coefficient n r. From the teams example aove we know that 5 3 =10: However, we know the numer of permutations of three people from the 5 possiilities is 5 4 3=60, so why is this answer 6 or 3! times larger than 5 3?

11 224 OUNTING ND THE INOMIL EXNSION (hapter 8) This can e seen if we consider one of these teams, say. There are 3! ways in which the memers of team can e placed in a definite order, i.e.,,,,,, and if this is done for all 10 possile teams we get all possile permutations of the 5 people taken 3 at a time. So, 5 4 3= 5 3 3! ) 5 3 = or 5! 3! 2!. n (n 1)(n 2)... (n r +3)(n r +2)(n r +1) In general, r n n! = r (r 1)(r 2) = {z } r!(n r)! {z } Factor form Factorial form Values of n r can e calculated from your calculator. For example, to find 10 3 : TI-83: ress 10 MTH 3 to select 3:n r from the R menu, then 3 ENTER asio: ress 10 OTN F6 F3 (RO) F3 (nr) 3 EXE Example 12 How many different teams of 4 can e selected from a squad of 7 if: a there are no restrictions the teams must include the captain? a There are 7 players up for selection and we want any 4 of them. This can e done in 4 7 =35ways. If the captain must e included and we need any 3 of the other 6, this can e done in =20ways. Example 13 committee of 4 is chosen from 7 men and 6 women. How many different committees can e chosen if: a there are no restrictions there must e 2 of each sex c at least one of each sex is needed? a For no restrictions there are 7+6=13 people up for selection and we want any 4 of them. ) total numer = 4 13 = 715: The 2 men can e chosen in 2 7 ways and the 2 women can e chosen in 2 6 ways. ) total numer = = 315: c Total numer = numer with (3 M and 1 W) or (2 M and 2 W) or (1 M and 3 W) = = 665 lternatively, total numer = : 6

12 OUNTING ND THE INOMIL EXNSION (hapter 8) 225 EXERISE 8E 1 Evaluate using factor form: a c 8 3 d 8 6 e 8 8: heck each answer using your calculator. 2 In question 1 you proaly noticed that 8 2 = 8 6. In general, r n = n r n. rove using factorial form that this statement is true. 3 Find k if 9 ³ k =4 7 k 1 : 4 List the different teams of 4 that can e chosen from a squad of 6 (named,,, D, E and F). heck that the formula for n r gives the total numer of teams. 5 How many different teams of 11 can e chosen from a squad of 17? 6 andidates for an examination are required to do 5 questions out of 9. In how many ways can this e done? If question 1 was compulsory, how many selections would e possile? 7 How many different committees of 3 can e selected from 13? How many of these committees consist of the president and 2 others? 8 How many different teams of 5 can e selected from a squad of 12? How many of these teams contain: a the captain and vice-captain exactly one of the captain or the vice-captain? 9 team of 9 is selected from a squad of of the players are certainties who must e included, and another must e excluded ecause of injury. In how many ways can this e done? 10 In how many ways can 4 people e selected from 10 if: a one person is always in the selection 2 are excluded from every selection c 1 is always included and 2 are always excluded? 11 committee of 5 is chosen from 10 men and 6 women. Determine the numer of ways of selecting the committee if: a there are no restrictions it must contain 3 men and 2 women c it must contain all men d it must contain at least 3 men e it must contain at least one of each sex. 12 committee of 5 is chosen from 6 doctors, 3 dentists and 7 others. Determine the numer of ways of selecting the committee if it is to contain: a 2 doctors and 1 dentist 2 doctors c at least one of the two professions. 13 How many diagonals has a 20-sided convex polygon? 14 There are 12 distinct points,,, D,..., L on a circle. Lines are drawn etween each pair of points. a How many lines i are there in total ii pass through? How many triangles i are determined y the lines ii have one vertex? 15 How many 4-digit numers can e constructed where the digits are in ascending order from left to right? Note: You cannot start with 0.

LEVEL I. 3. In how many ways 4 identical white balls and 6 identical black balls be arranged in a row so that no two white balls are together?

LEVEL I. 3. In how many ways 4 identical white balls and 6 identical black balls be arranged in a row so that no two white balls are together? LEVEL I 1. Three numbers are chosen from 1,, 3..., n. In how many ways can the numbers be chosen such that either maximum of these numbers is s or minimum of these numbers is r (r < s)?. Six candidates