Combination - Order does not matter
You can recognise if a question is a combination question if it specifically says 'order does not matter' or the question did not tell you in what order should you arrange the objects.
Listed below are solutions to GCE 'O' level past 10 years examination questions on combinations.
Eg 1,
A committee of 6 is chosen from 10 men & 7 women so as to contain at least 3 men & 2 women. How many ways can this be done.
The combinations are:
1) 4 men and 2 women, or
2) 3 men and 3 women
Under the probability rule, and is multiplication(x) and or is addition(+).
a) 4 men and 2 woman : 10C4 x 7C2 = 4,410
b) 3 men and 3 woman : 10C3 x 7C3 = 4,200
4,410 + 4,200 = 8,610
Eg 2,
There are 4 pigeon holes marked W, X, Y and Z. In how many ways can 14 different books be arranged so that 5 of them are in W, 4 of them are in X, 3 of them are in Y and the rest are in Z? The order of books in each hole does not matter.
hole W : 5 books, and
hole X : 4 books, and
hole Y : 3 books, and
hole Z : 2 books.
Total : 14 books
hole W(choose from 14 books) : 14C5 = 2,002 ways
hole X(14-5=9 books left to choose) : 9C4 = 126 ways
hole Y(9-4=5 books left to choose) : 5C3 = 10 ways
hole Z(5-3=2 books, nothing to choose) : 1
So, the total number of ways of arranging the books = 2,002 x 126 x 10 x 1
= 2,522,520
Eg 3
A committee of 7 members is to be selected from 8 men and 5 women. Calculate the number of ways in which this can be done if
a) there are no restrictions,
b) the committee is to have exactly 4 men,
c) the committee is to have at least 1 women,
d) the committee is to have more women than men.
a) 13C7 = 1,716
b) 8C4 x 5C3 = 700
c) All men committee : 8C7 = 8
1,716-8 = 1,708
d) The combinations are:
i) 3 men 4 women = 8C3 x 5C4 = 280
ii) 2 men 5 women = 8C2 x 5C5 = 28
Total : 280 + 28 = 308
Eg 4
A Club formed by some undergraduates intends to send a group of 10 members to attend an exhibition. The club has a total of 20 members, of which 8 are Arts undergraduates, 7 are Business undergraduates and 5 are Engineering undergraduates. Find the number of ways to choose the group if
a) the selection process is random,
b) 5 Arts, 3 Business and 2 Engineering undergraduates must be chosen
c) an equal number of Business and Engineering undergraduates must be chosen,
If the club intends to send a group comprising 5 Arts, 3 Business and 2 Engineering undergraduates, calculate the number of ways in which
d) the group may be chosen if it is to include 3 specified Arts and 1 specified Business undergraduates,
e) the 5 Arts undergraduates may be chosen from the 8 available, given that 2 particular Arts undergraduates cannot be chosen together.
20 memebers: 8 Arts + 7 Biz + 5 Eng
a) 20C10 = 184,756
b) 8C5 x 7C3 x 5C2 = 19,600
c) The combinations are:
8A+1B+1E, or
6A+2B+2E, or
4A+3B+3C, or
2A+4B+4C, or
0A+5B+5C
8C8 x 7C1 x 5C1 = 35
8C6 x 7C2 x 5C2 = 5880
8C4 x 7C3 x 5C3 = 24,500
8C2 x 7C4 x 5C4 = 4,900
7C5 x 5C5 = 21
Total = 35,315
d) 2A and 3A(specified) and 2B and 1B(specified) and 2E
5C2 x 6C2 x 5C2 = 1,500
e) no restriction in arts students : 8C5 = 56
if the 2 arts students were chosen together : 6C3 = 20
if the 2 arts students were not chosen together : 56 - 20 = 36
3B and 2E = 7C3 x 5C2 = 350
350 x 36 = 12,600
Eg 5 (N99/P2/24c)
An art gallery has 9 paintings by a famous artist. A selection of 4 of these are to be shown in an exhibition. Calculate the number of different selections that can be taken if (i) there are no restrictions, (ii) one special painting must be included.
(i) 9C4 = 126
(ii) 1 special painting and (choose the other 3 painting from the balance 8 paintings)
1 special painting = 1
choose 3 paintings from 8 paintings : 8C3 = 56
So, the number of selections are : 1 x 56 = 56
Eg 6 (N2002/P2/5a)
The producer of a play requires a total cast of 5, of which 3 are actors and 2 are actresses. He auditions 5 actors and 4 actresses for the cast. Find the total number of ways in which the cast can be obtained.
actors : 5C3 = 10, and
actresess : 4C2 = 6
So, the total number of ways the cast can be obtained are : 10 x 6 = 60
Eg 7 (N2003/92/8)
A garden centre sells 10 different varieties of rose bush. A gardener wishes to buy 6 rose bushes, all of different varieties.
(i) Calculate the number of ways she can make her selection.
Of the 10 varieties, 3 are pink, 5 are red and 2 are yellow. Calculate the number of ways in which her selection of 6 rose bushes could contain
(ii) no pink rose bush
(iii) at least one rose bush of each colour.
(i) 10C6 = 210
(ii) 10 rose : 3 pink + 5 red + 2 yellow
no pink rose, so left with 7 roses to choose for 6 rose : 7C6 = 7
(iii)
a) P P P R R Y, or
b) P P R R R Y, or
c) P R R R R Y, or
d) P R R R Y Y, or
e) P P R R Y Y, or
f) P P P R Y Y, or
a) 3C3 x 5C2 x 2C1 = 20
b) 3C2 x 5C3 x 2C1 = 60
c) 3C1 x 5C4 x 2C1 = 30
d) 3C1 x 5C3 x 2C2 = 30
e) 3C2 x 5C2 x 2C2 = 30
f) 3C3 x 5C1 x 2C2 = 5
Total = 175 ways
Eg 8 (N2004/P1/7b)
3 students are selected to form a chess team from a group of 5 girls and 3 boys. Find the number of possible teams that can be selected in which there are more girls than boys.
a) G G B, or
b) G G G
a) GGB : 5C2 x 3C1 = 30
b) GGG : 5C3 = 10
So the number of teams are 30 + 10 = 40
Eg 9 (N2005/P2/11a)
Each day a newsagent sells copies of 10 different newspapers, one of which is The Times. A customer buys 3 different newspapers. Calculate the number of ways the customer can select his newspapers. (i) if there is no restriction (ii) if 1 of the 3 newspapers is The Times.
(i) 10C3 = 120
(ii) The Times & choose 2 more other newspapers from the other 9 newspapers
The Times = 1
Choose 2 newspapers from 9: 9C2 = 36
So, the number of ways to select the newspapers are : 1 x 36 = 36 ways.
Eg 10 (N2006/P2/10)
a) How many different four-digit numbers can be formed from the digits 1,2,3,4,5,6,7,8,9 if no digit may be repeated?
b)In a group of 13 entertainers, 8 are singers and 5 are comedians. A concert is to be given by 5 of these entertainers. In the concert there must be at least 1 comedian and there must be more singers than comedians. Find the number of different ways that the 5 entertainers can be selected.
a) 9C1 x 8C1 x 7C1 x 6C1 = 3,024
b) 13 entertainers : 8 singers + 5 comedians
concert (5 entertainers) : >= 1 comedian, singers > comedians
The combinations are:
a) C S S S S, or
b) C C S S S
a) 5C1 and 8C4 = 350
b) 5C2 and 8C3 = 560
Total number of ways is 350 + 560 = 910.
Eg 11 (N2007/P2/8ai)
Each of seven cards has on it one of the digits 1,2,3,4,5,6,7; no two cards have the same digit. Four of these cards are selected and arranged to form a 4-digit number. (i) How many different 4-digit numbers can be formed in this way?
1st number(choose 1 from 7 numbers) : 7C1 = 7
2nd number(choose 1 from balance 6 numbers): 6C1 = 6
3rd number(choose 1 from balance 5 numbers): 5C1 = 5
4th number(choose 1 from balance 4 numbers): 4C1 = 4
Total number of ways is 7 x 6 x 5 x 4 = 840
Eg 12 (N2007/P2/8b)
4 people are selected to form a debating team from a group of 5 men and 2 women.
(i) Find the number of possible teams that can be selected.
(ii) How many of these teams contain at least 1 woman?
(i) 7C4 = 35
(ii) all men team : 5C4 = 5
35 - 5 = 30
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