When do I use the multiplication principle?

Multiply the choices at each step; with four letters and three boxes that is 3 to the power 4.

What should I watch when bundling?

Arrange the bundle as one unit, then multiply by the internal orders, not just once.

Why divide by factorials for repeated letters?

Swapping identical letters makes no new arrangement, so divide by the factorial of each repeat count.

S5 Mathematics: Permutations and Combinations Practice Paper 4

2026-11-03

Tutor Prep Guide · Secondary Mathematics

Set four closes the chapter with new numbers: seven in a line, two of eleven, two posts among nine and four letters into three boxes, then twelve choose three, three people together and at least one boy, then five couples adjacent and the letters of MATHEMATICS, ending with the error of ignoring which post is which.

Quick answer　5 040, 55, 72, 81, 220, 144, 74, 3 840, 4 989 600 and 20.

01　Paper details (check answers afterwards)

Item	Detail
Questions	10 questions
Suggested time	25 minutes
Difficulty mix	Foundation 4 · Intermediate 3 · Challenge 2 · Error redo 1
Chapter	Chapter: S5 Mathematics: Permutations and Combinations – Tutor Lesson Plan
Level	S5

02　Questions (10 items)

FoundationSeven students stand in a line. How many different orders are possible?
FoundationTwo class representatives are chosen from 11 students with no separate posts. How many ways?
FoundationFrom 9 students, choose a chairperson and a vice-chairperson who must be different people. How many outcomes?
FoundationFour different letters are posted into 3 different boxes, any number per box. How many ways?
IntermediateHow many 3-person groups can be formed from 12 students?
IntermediateSix people queue up with A, B and C standing together. How many orders are possible?
IntermediateFrom 4 boys and 5 girls, choose 3 people including at least one boy. How many combinations?
ChallengeFive married couples stand in a line with each couple adjacent. How many orders?
ChallengeHow many distinct arrangements are there of all 11 letters in MATHEMATICS?
Error redoTwo of five people are chosen as leader and recorder. A student answers 10. What is correct?

03　Answers and one-line notes

#	Answer	One-line note
1	5 040 orders	Seven factorial, that is 7 x 6 x 5 x 4 x 3 x 2 x 1, gives 5 040.
2	55 ways	Order does not matter, so C(11,2) is 11 x 10 divided by 2, giving 55.
3	72 outcomes	The two posts are distinct, so 9 x 8 gives 72.
4	81 ways	Each letter has 3 choices, so 3 to the power 4 gives 81.
5	220 groups	C(12,3) is 12 x 11 x 10 divided by 6, giving 220.
6	144 orders	Bundle the three together: 3! x 4! is 6 x 24 = 144.
7	74 combinations	Take all C(9,3) = 84 and subtract the all-girl C(5,3) = 10, leaving 74.
8	3 840 orders	Each couple has 2 internal orders, so 2 to the power 5 times 5! is 32 x 120 = 3 840.
9	4 989 600 arrangements	M, A and T each appear twice, so divide 11! by 2! x 2! x 2!, giving 4 989 600.
10	20 outcomes	The posts differ so use P: 5 x 4 = 20, whereas 10 is C(5,2) and ignores which post is which.

04　What to do after marking

After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
Repeat the same item two days later; move to the next paper only when both attempts are correct.
Parents only need the answer and note columns; there is no need to explain every item.

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