Why are five people arranged in 120 ways?

The first place has 5 choices and each later place one fewer, so 5 x 4 x 3 x 2 x 1 gives 120.

Why is the three-post choice not a combination?

The posts differ, so swapping people changes the outcome and order must be counted.

Why divide by factorials for repeated letters?

Swapping identical letters gives the same arrangement, so divide by the factorial of each count.

S5 Mathematics: Permutations and Combinations Practice Paper 2

2026-11-03

Tutor Prep Guide · Secondary Mathematics

Set two uses fresh numbers: arrangements of five people, choosing three of seven and three distinct posts to fix factorial against C(n,r), then ten choose four, mixed groups and a block method, then two unlabelled groups and the repeated letters of MISSISSIPPI, ending with the error of answering 21 instead of 42.

Quick answer　120, 35, 336, 24, 210, 36, 240, 126, 34 650 and 42.

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)

FoundationFive students stand in a line. How many different orders are possible?
FoundationHow many ways are there to choose 3 books from 7 different books?
FoundationFrom 8 people, pick a chairperson, a vice-chairperson and a secretary, all different. How many outcomes?
FoundationHow many arrangements are there of the four letters A, B, C and D?
IntermediateHow many 4-person teams can be formed from 10 students?
IntermediateFrom 4 boys and 4 girls, choose 2 boys and 2 girls. How many combinations?
IntermediateSix people queue up with A and B standing next to each other. How many orders are possible?
ChallengeTen people are split into two groups of 5, with the groups unlabelled. How many ways?
ChallengeHow many distinct arrangements are there of all 11 letters in MISSISSIPPI?
Error redoFrom 7 people, choose 2 where one becomes class monitor and one becomes deputy. How many arrangements?

03　Answers and one-line notes

#	Answer	One-line note
1	120 orders	Multiply 5 x 4 x 3 x 2 x 1, which is 5 factorial, giving 120.
2	35 ways	Order does not matter here, so use C(7,3), which is 7 x 6 x 5 divided by 6, giving 35.
3	336 outcomes	That is 8 x 7 x 6, because the three roles differ and order matters.
4	24 arrangements	It is 4 factorial, that is 4 x 3 x 2 x 1 = 24.
5	210 teams	C(10,4) is 10 x 9 x 8 x 7 divided by 24, giving 210.
6	36 combinations	Take C(4,2) times C(4,2), which is 6 x 6 = 36.
7	240 orders	Treat A and B as one block so there are 5! orders, times 2 for swapping them, giving 240.
8	126 ways	That is C(10,5) divided by 2!, which is 252 divided by 2, giving 126.
9	34 650 arrangements	Divide 11! by 4! x 4! x 2!, which is 39 916 800 divided by 1 152, giving 34 650.
10	42 arrangements	The posts differ, so order matters and the answer is 42, not the 21 of C(7,2).

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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