School of Engineering
A113 – Mathematics
Collaborative Learning Worksheet for Problem 10: Game Plan
Practice Questions
(It is essential to complete these practice questions so that you can understand the concepts of
this lesson better and be more confident and competent in handling related questions.)
Give non-exact numerical answers correct to 2 decimal places unless a different
level of accuracy is specified in the question.
The Factorial Function
1. Without using a calculator, calculate the value of
12 !
.
10 !
132
Arrangement without repetition
[You may want to watch the following video which would help to recap some of the key concepts learnt
before attempting the following question.]
2. There are one pink, one purple, one indigo and one white marble to be placed in four
different containers C1, C2, C3 and C4.
If every container is to contain only 1 marble, how many different ways can we
arrange the marbles in the different containers?
24
3. How many ways can you arrange the letters in the word TEA?
6
Arrangement with repetitions
4. How many ways can you arrange the letters in the word COFFEE?
720
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5. There are two identical purple and three identical pink marbles to be placed in five
different containers C1, C2, C3, C4 and C5. If every container is to contain only 1
marble, how many different ways can we arrange the marbles into the different
containers?
10
Arrangement with constraints / groups
6. There are one pink, one indigo, one grey, one white and one purple marble to be
placed in five different containers C1, C2, C3, C4 and C5. If every container is to
contain only 1 marble, how many different ways can we arrange the marbles into the
different containers with the following conditions?
a) The pink marble has to be placed in container C1.
24
b) The indigo and grey marbles have to be placed in containers next to each other,
while the pink, white and purple marbles have to be placed in containers next to
one another.
2
7. Seven letters (A, B, C, D, E, F, G) are arranged in a line.
a) How many different arrangements are possible?
5040
b) How many different arrangements are possible if the 2nd letter in the line must be
C?
720
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Combination and Permutation
[You may want to watch the following video which would help to recap some of the key concepts learnt
before attempting the following question.]
8. Five coloured balls (pink, grey, indigo, purple and white) are placed in a bag.
How many ways are there to choose 3 balls from the bag?
10
9. Five coloured balls (pink, grey, indigo, purple and white) are placed in a bag.
If we randomly pick out 3 balls from the bag and place them in different containers
C1, C2 and C3, how many different ways can we arrange the balls?
20
10. How many 3-digit numbers can be formed from the digits 1, 2, 3, 4, 5, 6, 7, 8, with
none of the digits repeated?
56
11. There are 10 Year-one students, 5 Year-two students and 4 Year-three students in a
CCA club. The club has to choose 4 representatives for a competition.
a) How many ways are there to choose a group of 4 students to participate in the
competition?
3876
b) How many ways are there to choose 2 Year-one students and 2 Year Two
students to participate in the competition?
55
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12. There are 3 boys and 3 girls.
a) How many ways are there to arrange them in a line?
72
b) If the 3 boys have to stand together, how many ways are there to arrange the 3
boys and 3 girls in a line?
6
c) If the 3 boys have to stand together and the 3 girls have to stand together, how
many ways are there to arrange the 3 boys and 3 girls in a line?
2
13. In how many ways can a group of 5 men and 2 women be chosen out of a total of 7
men and 3 women?
63
14. Consider the word APPLE.
a) How many ways can you arrange the letters in the word APPLE?
60
b) If the word is changed to PINEAPPLE, how many more ways are there to
arrange the letters compared to part (a)?
30180
c) If the word is changed to PINEAPPLE, how many ways are there to arrange the
letters if all 3 P’s cannot be grouped together?
27720
15. There are 20 competitors in a swimming contest.
How many ways can the first, second and third positions be taken up by the
competitors?
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1140
16. In a clothes store, pants with the same design come in 3 colours: green, blue and
yellow. Skirts with the same design come in 4 colours: yellow, purple, red and green.
A lady wants to buy a pair of pants for her son and a skirt for each of her two
daughters. Assuming all her children wear clothes of different sizes, how many
different ways can she make her purchase if the pants and skirts must all be different
in colour?
17. There are 5 books on English, 2 books on Accounting, 4 books on Mathematics and
3 books on Sports. If we have to keep books of a subject together, in how many ways
can we arrange them in a single row of the shelf?
829440
18. There are 8 different routes for Peter to travel between City A and City B. In how
many ways can Peter travel from City A to City B and travel back to City A without
using the same route in the return journey?
56
19. There are 6 male students and 6 female students. How many ways are there to
arrange all the students in a row if there should not be any student of the same
gender standing together?
1036800
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Solving Scenario
Putting it together
1. Kobe Bryant, being a star player, needs to be in your team for the match
against Shaq.
Based on what you had learnt,
a) Calculate the number of combinations of the team that you can form for the
match.
4C x 2C x 2C
1
1
1
x 4C1 = 64
b) Calculate the number of ways to choose any 2 reserves from the remaining
players left in the roster.
11C
2
= 55
c) How many possible ways are there to assign a captain and a co-captain in
the team?
Exploring further
2. Suppose there is another match against a low-ranked team, and you can
choose to omit Kobe Bryant in your team line-up. It has also come to your
attention that Kobe is in a dispute with Andrew Bynum and therefore the two
may not work well together. Hence, if Kobe is in the team, you like to omit
Andrew and likewise, if Andrew is in the team, you like to omit Kobe.
Determine the number of ways which you can form the team for this match.
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