DSE Maths: Permutations and Combinations, a Tutor Prep Guide

2026-11-04

Tutor lesson-prep guides · Secondary Maths

Students pick C for anything that says choose and P for anything that says arrange, then collapse on the restriction. This lesson teaches two decisions only: does swapping change the result, and can the complement do the work.

Objectives — Students can: (1) decide between P and C with one sentence about swapping; (2) handle at least one type restrictions with the complement method; (3) separate grouping (unordered) from distributing (ordered); (4) in five minutes count the ways to choose 4 from 9 and the ways with at least one girl (126 and 121).

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Factorial	What is 4 factorial?	24
P and C	C(5,2) and P(5,2)?	10 and 20
Complement	What is the opposite of at least one?	None at all

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-8 min	Three-question check	Factorial, P and C, complement	Answer orally
8-20 min	Decide the order	Three situations, swap or not	Say P or C
20-34 min	Restrictions	Complement method against direct counting	Recount by complement
34-46 min	Group and distribute	Group first, then arrange if the groups are named	Tell the two apart
46-56 min	Timed practice	Three questions, easy to hard	Work unaided
56-60 min	Exit check	Three checkpoints, one question each	Self-mark and circle slips

03 Teaching script (read it out)

Scene 1: One sentence decides P or C

Ask whether swapping the two people gives a different result. Different means P; the same means C. Choosing a group that does one job is C, choosing people for different posts is P. This decision has to be made within seconds of reading, because every later line depends on it. Get it backwards and perfect arithmetic still scores zero.

Each question begins with a spoken sentence: swapping matters, so P, or swapping does not, so C.

Scene 2: Restrictions: count the side you are allowed to subtract

At least one girl directly means one girl, two, three or four, and students lose cases. Use the complement: the total is C(9,4) = 126, subtract the all-boy case C(5,4) = 5, leaving 121. Two lines instead of four cases. The rule of thumb: if the matching cases outnumber the failures, subtract them instead of adding the matches.

Write total minus complement on paper with both numbers; no mental arithmetic.

Scene 3: Group, then distribute if the groups are named

Splitting people into unnamed groups is C; sending those groups to named places adds an arrangement. Choosing 2 boys from 5 and 2 girls from 4 is C(5,2) times C(4,2), which is 10 times 6, so 60. If the two pairs then go to two different rooms, multiply by two because the rooms differ. A circular arrangement is (n – 1) factorial, so six people round a table give 120, since rotations count as the same.

Circle the words that name the groups; if the groups are not named, do not multiply by their number.

04 Question types and worked steps

Type	Example	Steps	Answer
Pure combination	Choose people for one job	Swapping unchanged, so C	C(9,4) = 126
Pure permutation	Choose people for different posts	Swapping changes, so P	P(9,3) = 504
Restriction	At least one girl	Total minus all-boy	121
Circular	Seated round a table	(n – 1) factorial, check for direction	120 ways

Sanity check (teach this): Magnitudes must behave: C(9,4) is 126 while two boys and two girls gives only 60, smaller because of the extra condition. At least one girl, 121, sits close to the total since only 5 all-boy cases are removed; that ratio is an instant sanity check. Permutations always exceed combinations: P(9,3) is 504 against C(9,3) of 84, a factor of six. A combination larger than a permutation means the formulas were swapped.

05 Independent practice (3 questions)

Question	Answer	What to watch
Choose 4 from 9 for a committee	126 ways	C chosen deliberately
Choose 4 from 9 with at least one girl	121 ways	Complement used
Choose 2 boys from 5 and 2 girls from 4	60 ways	Two steps multiplied

06 Common mistakes and what to say

Mistake	Student writes	Say this
P and C swapped	Choosing a committee with P, giving 3024	A committee does the same job either way, so it is C, not P.
Counting at least cases one by one	One girl only, missing three and four	Subtract the all-boy case from the total instead.
Forgetting named groups	Only C when filling two rooms	The rooms differ, so multiply by two after grouping.
Circle treated as a line	Six people written as 6 factorial	Rotations are identical, so it is (6 – 1) factorial.

07 Homework (paper, section and question numbers)

Maths subject index — All chapters
Practice paper — Counting: questions 5 and 8

08 Exit check (last 5 minutes)

Question	Answer	Decision
P or C	Does swapping change the result	Cannot say: reteach act one
Complement	total minus the opposite	Counting cases: redo the example
Groups	group then arrange	Missing factor: act three again

09 Tutor FAQ

Why is 0 factorial 1?

There is exactly one way to arrange nothing, and defining it as 1 keeps the formulas consistent.

Does at least two also use the complement?

Yes, but the opposite now has two cases, zero and one, and both must be subtracted.

What about repeated items?

Divide n factorial by the factorial of each repeated group, so AAB gives 3 over 2, which is 3.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.