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Topic : COMBINATIONS, BINOMIAL THEOREM TIME : 4 X 45 minutes

STANDARD COMPETENCY :

1. To use the statistics rules, the rules of counting, and the characteristic of probability in problem solving.

BASIC COMPETENCY:

1.5 To use the rules of multiplication , permutation, and combination in problem solving. In this chapter, you will learn about: • To arrange the rules of combination of r objects from n different objects. • To use the rules of combination.

• To use the rules of Binomial Newton to determine the expansion ofi ( )nba +

D. COMBINATIONS

Having studied permutations where the order of each object is important, let us turn our attention to combinations. A combination is any selection of objects where the order of the objects is immaterial (of no concern) For example, the different (ordered) permutations ABC and CAB are considered as the same combination when we disregard the order of the letters and realize that both contain the same three letters. Consider the permutations of 2 people from 4 people, P,Q,R, and S. We have learned that there are 4P2 = 12 such permutations.

Permutations (order important)

Combinations (order not important)

∴ The combinations of 2 people from 4 people are … combinations

PQ QP PQ PR RP PR PS SP PS QR RQ QR QS SQ QS RS SR RS

Consider the permutations of 3 people from 4 people, P,Q,R, and S. We have learnt that there are ….P…. = ….. such permutations.

Worksheet 3Worksheet 3Worksheet 3Worksheet 3rdrdrdrd

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Permutations (order important)

Combinations (order not important)

∴ The combinations of 3 people from 4 people are … combinations

PQR QPR RQP PRQ QRP RPQ PQR PSQ ….. ….. ….. .…. ….. …… PRS ….. ….. ….. .…. ….. …… QRS ….. ….. ….. .…. ….. ……

In general, the number of combinations of r objects from n different objects is

( ) !!!

rCC rn

nr

KK

K

−==

Example 19 From 8 students shall be chosen 3 students for a visit to Jakarta. Find the number of possibilities in choosing the three students.

Solution

( ) KKK

KKKK

KKK

K

KK=×××=

−=

!!!

!!!

C

Example 20 A committee of 5 members is to be selected from 6 seniors and 4 juniors. Find the number of ways in which this can be done if a. there are no restrictions, b. the committee has exactly 3 seniors, c. the committee has at least 1 junior.

Solution a. If there are no restrictions, the 5 members can be selected from …. people in

( ) KKKKK

KKK

KK

KKKKKK

KK

K

KKK

K

KK==×××××==

−=

!!!

!!!

!!!

C ways

b. If the committee has exactly 3 seniors, these seniors can be selected from … seniors

in …C… ways. The remaining 2 members must then be juniors which can be selected from … juniors in …C… ways. By the basic counting principle, the number of committees with exactly 3 seniors is: …C… x …C… = ….

c. Number of committees with no junior (it means that all 5 members are seniors)

= …C… x …C… = ….

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So the number of committees with at least 1 junior = total number of committees – number of committees with no junior = …... - …… = ……

Exercise 4 1. Evaluate 5C4 and 9C3.

2. Using the result ( ) !!!

rrn

nCrn −

=

a. evaluate nC0 and nCn , b. find nC1 and nC2 in terms of n, c. prove that nCr = nCn-r

3. In a soccer league consisting of twelve sides, each team plays every other team once. How many matches are there?

4. A club has 14 members. It has to send a delegation of 5 members to represent them at a

particular event. Find the number of possible delegations.

5. At the library, Ina found 6 books of interest but he can only borrow 4 books. How many possible selections can he make?

6. The second section of Mathematics paper contains 7 questions and a candidate must

answer any 4 questions. In how many ways can the 4 questions be chosen (without regard to order)?

7. On a piece of paper, 8 points are marked such that no 3 points lie on the same straight

line. How many lines can be drawn passing through any 2 of these points? 8. Seven points lie on a circle. How many triangles can be drawn using any 3 of these points

as vertices? 9. A group of 4 adults and 3 children are to be formed from 8 adults and 5 children. How

many possible groups are there? 10. To promote reading, a teacher decides to feature 3 classics, 4 contemporary novels and 2

non-fiction books on a notice board. How many selections can she make from 5 classics, 6 contemporary novels and 4 non-fiction books?

11. Calculate the number of ways in which

a. 5 children can be divided into groups of 2 and 3 b. 9 children can be devided into groups of 5 and 4. Hence calculate the number of ways in which 9 children can be devided into groups of 2, 3 and 4.

12. For a community service project, 10 students volunteer to do some cleaning up at an old folks’ home. The teacher-in-charge wants to split them into 3 groups. One group will consist of 5 students to clean the rooms; another group of 3 students to clean the toilets;

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the last group of 2 students to sweep the corridors. Find the number of ways the teacher-in-charge can form the groups.

13. A club contains the President, the Secretary and 8 other members. The club is asked to

send a group of 4 representatives to a conference. Calculate the number of different ways in which the group can be formed if it must contain a. both the President and the Secretary b. either the President or the Secretary, but not both.

14. A particular firm has 6 vacancies to fill from 15 applicants. Calculate the number of ways in which these vacancies could be filled if there are no restrictions. The firm decides that 3 of the 6 vacancies shall be filled by women and 3 by men. The applicants consist of 7 women and 8 men. Calculate the number of ways in which 6 vacancies could be filled under these conditions. One of the 7 women is the wife of one of 8 men. Calculate the number of ways in which 3 women and 3 men could fill these 6 vacancies, given that both the wife and her husband are among those appointed.

15. A delegation of 4 people is to be selected from 5 women and 6 men. Find the number of possible delegations if a. there are no restrictions, b. there is at least 1 woman, c. there are at least 2 women. One of the men cannot get along with one of the women. Find the number of delegations which include this particular man or woman, but not both.

16. A chess club has 10 members, of whom 6 are men and 4 women. A team of 4 members is selected to play in a match. Find the number of different ways of selecting the team if a. all the prayers are to be of the same sex, b. there must be an equal number of men and women. Given that the 6 men include 2 brothers, find the total number of ways in which the team can be selected if either of the brothers, but not both, must be included.

17. Five different letters are chosen from the letters A,B,C,D,E,F,G,H,I, and J. Find the number of choices. How many of these choices contain a. exactly 3 consonants, b. no vowel, c. at least one vowel?

18. An art gallery has a total of 11 paintings by a particular artist; of these, 5 are oil paintings and 6 are watercolour paintings. The art gallery is asked to provide a special exhibition of this artist’s work but is restricted to showing only 5 paintings. Calculate the number of ways in which the 5 paintings can be selected for the exhibition if a. there are no restrictions, b. one particular painting must be included, c. there must be exactly 3 watercolour paintings, d. there must be more watercolour paintings than oil paintings.

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Example 21 2 Mathematics books, 4 Science books and 3 Literature books are to be arranged on shelf. In how many ways can this be done if a. there are no restrictions? b. the 2 Mathematics books are not together? c. books of the same subject must be placed next to one another? If 5 of these 9 books are to be transferred and arranged on another shelf, in how many ways can this be done if the first and last books on this shelf are Literature books?

Solution a. If there are no restrictions, number of ways = …! = ….. b. Gather these 2 books together to form one block. Within this block, these 2 books can

be permuted in …! ways. This block and the other 7 books then form … objects which can be permuted in …! ways. So the number of such arrangements = …! x …! = …….

∴ number of ways in which the 2 Mathematics books are not together = ……… - ………. = ……….. c. We gather books of the same subject together. This results in … blocks which can be

arranged in …! ways. Within the bock of 2 Mathematics books, there are …! permutations. Within the bock of 4 Science books, there are …! permutations. Within the bock of 3 Literature books, there are …! permutations. Hence by the basic counting principle, the required number of ways = …! x …! x …! x …! = ……

5 books are to go to another shelf. The 1st place can be filled by any of 3 Literature books. The last place can be filled by any

of the remaining 2 Literature books. Thus far, we have

The remaining 3 places can be filled by 3 of the remaining 7 books in …P… ways. ∴ the required number of ways = … x … x …P… = ………

Example 22 A committee of 5 members are to be formed from 5 married couples. Find the number of committees if a. the selection is random (there are no restrictions), b. a particular couple is in the committee, c. there are more men than women.

Solution

Miscellaneous Examples

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a. b. c. (Miscellaneous) Exercise 5 1. Tomi wishes to contribute three items to a charity gift box. He narrows his choices to

seven particular items. Find the number of ways he can select the three items from these seven items.

2. How many 5-letter words ending with a consonant can be formed using letters from the

word EDUCATION? 3. A book store’s bestseller list shows its 10 most popular books. Romi wishes to buy 4 of

them as gifts. How many possibilities are there? 4. Eight friends are to make a journey in 2 taxis, with 4 people in each taxi. In how many

ways can this be done? 5. A particular class consists of 25 Chinese, 6 Malays, 4 Indians, and 3 Eurasians. Two

students from each race are to be selected for a Racial Harmony Day concert item. In how many ways can this be done?

6. There are 6 males and 6 females in the finals of a talent competition. A contest is held to

pick the top 3 winners in both the male and female categories in order of merit. How many different entries must be complicated to ensure a winning order?

7. A certain contest awards three top prizes of $50 000, $20 000, and $10 000 and five

consolation prizes of $1000 each. Twenty people have each submitted one winning entry and a draw is carried out to pick the eight different winners. Firstly, the five winners for the consolation prizes are drawn. How many outcomes are possible? Next, the three winners for the top prizes are drawn. How many possible results are there?

8. Find how many numbers between 2500 and 5000 can be formed using digits from 1, 2, 3, 4, 5, and 7, with no digit being repeated.

9. Find the number of ways 3 boys and 4 girls can be arranged in a row if

a. there are no restrictions b. a particular boy and a particular girl must be next to each other, c. the first and last persons are of the opposite sex.

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10. Calculate in how many ways each of the following choices can be made:

a. 4 books are to be chosen from a list of 9 titles to take away for reading during a holiday,

b. 15 people have sent in winning entries for a magazine competition, and three are to be chosen and placed in order of merit so as to receive the 1st, 2nd, and 3rd prizes.

c. A committee of 3 people comprising the president, secretary, and treasurer are to be chosen from 8 possible candidates.

11. Find the total number of arrangements using all of the letters in the words LOGARITHMS.

How many of these arrangements a. start with H and end with A? b. have three consecutive vowels? c. have the consonants G and H separated?

12. A committee of 6 members is to be chosen from 6 women and 5 men. Calculate the number of ways this can be done if a. the selection is random, b. a minimum of 4 men must be chosen.

13. A tennis team of 4 men and 4 women is to be picked from 6 men and 7 women. Find the number of ways in which this can be done. It was decided that 2 of the 7 women must either be selected together or not at all. Find how many possible teams could be selected in these circumstances. This selected team is arranged into 4 pairs, each consisting of a man and a woman. Find the number of ways in which this can be done.

14. From a group of 6 girls and 7 boys, how many 5-member committee consist of a. 3 girls and 2 boys? b. 3 boys and 2 girls? c. members of the same sex? d. more boys than girls?

15. Find the number of ways in which a team of 6 batsmen, 4 bowlers and a wicket-keeper may be selected from a squad of 8 batsmen, 6 bowlers, and 2 wicket-keepers. Find the number of ways in which a. this team may be selected if it is to include 4 specified batsmen and 2 specified

bowlers, b. the 6 batsmen may be selected from the 8 available, given that 2 particular batsmen

cannot be selected together.

16. Which of the following statements are incorrect? Give your reasons. a. Group A and Group B are to be formed from 7 people. Group A is to have 4 people

and Group B is to have 3 people. This can be done in 7C3 ways. b. From a set of 6 distinct elements, 6P4 subsets of 4 elements can be formed. c. 9 people are to form three queues of sizes 2, 3, and 4. This can be done in 9C2 x 7C3 x

4C4 ways. d. 4P3 different codes can be formed from the letters in the word FREE.

17. The number of applicants for a job is 15. Calculate the number of ways in which 6 applicants can be selected for interview. The 6 selected applicants are interviewed on a particular day. Calculate the number of ways in which the order of the 6 interviews can be arranged. Of the 6 applicants, 3 have backgrounds in business, 2 have backgrounds in education and 1 has a background in recreation. Calculate the number of ways in which the order of

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6 interviews can be arranged, when applicants having the same background are interviewed in succession.

18. Calculate the number of ways of selecting 2 points from 6 distinct points. Six distinct points are marked on each of two parallel lines. Calculate the number of a. distinct quadrilaterals which may be formed using 4 of 12 points as vertices, b. distinct triangles which may be formed using 3 of the 12 points as vertices.

19. A shelf is to contain 7 different books, of which 4 were written by Dicky and 3 by Hary. Find the number of arrangements in which a. no two books by the same author are adjacent, b. the first two books at the left-hand end are by the same author.

20. A concert pianist agrees to take part in a charity concert. She offers a choice of 10 works, of which 5 were composed by Chopin, 3 by Liszt, and 2 by Schumann. Calculate the number of ways these 4 works can be selected if a. there are no restrictions, b. there are 2 works by Chopin, 1 by Liszt, and 1 by Schumann, c. there must be at least one work by each composer, d. the selection contains works by exactly two of the three composers.

E. BINOMIAL THEOREM Expression such as )( yx + , )1( x− , and )( ba + which contain two terms each, are called

binomials. Let us take a binomial )( ba + . If we keep on multiplying )( ba + by itself, we get

)( ba + , 2)( ba + , 3)( ba + , 4)( ba + , 5)( ba + , … We list the following expansions of nba )( + .

0)( ba + = 1

1)( ba + = … + …

2)( ba + = ….. + ….. + …..

3)( ba + = ….. + ….. + ….. + …..

4)( ba + = ….. + ….. + ….. + ….. + …..

5)( ba + = ….. + ….. + ….. + ….. + ….. + …..

Based on the expansions above, we draw the following rules concerning the expansions of

nba )( + . a. There are )( ba + terms.

b. The power of b starts with 0 and increases to n , i.e. its powers are in ascending order.

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We list the coefficients, which are known as binomial coefficients. The Pascal’s triangle becomes:

1=n 1 + 1 01C 11C

2=n 1 + 2 + 1 02 C 12C 22 C

3=n 1 + 3 + 3 + 1 ….. …… …… …… 4=n … + … + … + … + … ….. …… …… …… ….. 5=n … + … + … + … + … + … ….. …… …… …… ….. …..

Hence the binomial expansion of nba )( + , where +Ζ∈n , is given by: n

nnn

nn bCaCba ..)( 0 ++++=+ KKKKKKK

KKK∑=

=+n

rrn

n Cba0

)(

Example 23 Expand 3)( ba + and hence find the expansion of 3)32( x+ .

Solution

3)( ba + = ∑K

KK

KKKK

= ………………………………….. = ………………………………….. Let 2=a and xb 3= .

Then, 3)32( x+ = ∑K

KK

KKKK

= …………………………………. = …………………………………. Example 24 Find, in descending powers of x , the first 4 terms in the binomial expansion of

a. 5)32( −x

b. 6

2

1

+x

x

Solution

a. 5)32( −x = ∑K

KK

KKKK

= …………………………………. = ………………………………….

= …………………………………. = ………………………………….

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b. 6

2

1

+x

x = ∑K

KK

KKKK

= …………………………………. = ………………………………….

= …………………………………. Example 25 Given that ( ) K++−=− 26

21 96 sxxrxp , find ,,rp and s .

Solution

( )621 xp − = ( )6

21 )( xp −+

= ∑K

KK

KKKK

= …………………………………. = ………………………………….

∴ 296 sxxr +−=++ KKKKK Comparing coefficients:

96:1 −=KKx ⇒ …… = 32 ⇒ p = ….

KKK ==rx :0

KKK ==sx :2 Example 26 In each of the following expansions, find the indicated term.

a.

112

22

+ x

, 7th term

b. 12

2

12

−x

x , term in 15

1x

Solution

a.

112

22

+ x

= ∑K

KK

KKKK

( ) KKKKKK =−==== rban ,,,

7th term: ( )K

K

K

=

2.2.

2

117

xCT

= ……………….. = ………………..

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b. KKKK

K

K

∑=

−+=

−12

2

12

2

12

12

xx

xx

KKK === ban ,,

Power of x in 12

2

12

−x

x = … - … - …

= … - …

For 15

1x

, so … - … = –15

..… = …. r = ….

∴ the term in 15

1x

= ( ) ( )KK

KKK.2. xC = …….

Exercise 6 01. Expand the following:

a. ( )423 x− b. ( )522 x+ c. 6

2

12

−x

02. Find, in ascending powers of x , the first 4 terms in the expansion of

a. ( )632 x+ b. 5

2

14

− x c. 8

222

1

− xx

03. Obtain, in ascending powers of x , the first 4 terms in the expansion of 7

22

− x.

Hence find the value of (1.995)7 correct to 4 decimal places.

04. Find the non-zero values of a and b for which ( ) 3233

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82 abxbxxax −+−=− .

05. Find the first 3 terms, in ascending powers of x , in the expansion of ( )921 x− and of

( )52 x+ . Hence expand ( ) ( )59 221 xx +− up to the terms in 2x . 06. Find the indicated term in each of the following expansions.

a. ( )102 x+ , 7th term c. ( )923 −x , 4th term

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b. ( )102xy − , 5th term d. 12

22

1

+x

x , middle term.

07. In the expansion of 10

23 2

−x

x , find

a. the term in 10x , b. the coefficient of 5

1x

, c. the constant term.

08. Expand 5

22

1

− x up to the term in 3x . If the coefficient of 2x in the expansion of

( )5

2 22

131

−++ xxax is 213 , find the coefficient of 3x .

09. In the expansion of ( )( )121 bxax −+ , where 0≠ab , the coefficient of 8x is zero. Find in its

simplest form the value of the ratio b

a.