DSE Maths | Permutations and Combinations Tutor Prep Guide

DSE Maths | Permutations and Combinations Tutor Prep Guide | nPr, nCr, restrictions

2026-11-01

Tutor lesson-prep guides · Secondary Maths

Students reach for a formula the moment they see “how many ways” and the usual slip is treating an unordered pick as ordered, which inflates the answer. The fix is one opening question: does swapping the order create a new case? This lesson makes that the first move for every question.

Objectives — Students can: (1) decide in one sentence whether order matters; (2) choose nPr or nCr and write the expression; (3) handle “a named person must be included”, “must be adjacent” and “must not be adjacent”; (4) divide out repeated counting.

01 Prerequisites (3-minute check)

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

Skill	Quick check	Pass mark
Factorials	meaning and simplification of n!	5! = 120, n!/(n-2)! = n(n-1)
Multiplication principle	steps multiply	3 shirts x 4 trousers = 12 outfits
Addition principle	cases add	two exclusive cases counted separately then added

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-10 min	Ordered or not	one question decides it	does swapping make a new case?
10-25 min	Two formulas	the link between nPr and nCr	P(10,3)=720, C(10,3)=120
25-38 min	Restrictions	place the restricted item first	pick 3 of 10 with A included
38-50 min	Repeated items	divide by repeats	divide by the factorial of repeats
50-56 min	Mixed questions	multiply steps, add cases	list every case
56-60 min	Exit check	collect and mark three items	done independently

03 Teaching script (read it out)

Scene 1: First move: does order matter?

Choosing 3 representatives from 10 students and lining 3 of them up for a photo are different questions. The line-up is ordered: P(10,3) = 720. The representatives are not: C(10,3) = 120. The gap is exactly 3! = 6, and 720 / 6 = 120.
So every question starts with one written line: does order matter? Only then pick up the calculator.Ask the student: choosing a chairperson, a vice-chairperson and a secretary – three posts or three people?

Student task: write both expressions for “choose 2 of 5” and “arrange 2 of 5 in seats” and compare.

Scene 2: Two formulas only, and the link between them

nPr = n! / (n-r)!, nCr = n! / [r!(n-r)!], so nCr = nPr / r!. Example: C(13,5) = 1287, P(7,2) = 42.
Do not make students memorise three formulas. Memorise one move: count as if ordered, then decide whether to divide by r!. That also covers grouping-then-arranging questions: group first, divide by the factorial of the number of groups, then multiply by arrangements inside groups.Reminder: use symmetry C(n,r) = C(n,n-r) as a check, since C(13,5) = C(13,8) = 1287.

Student task: how many times larger is P(10,3) than C(10,3)? (Six times, that is 3!.)

Scene 3: Restrictions: place the awkward item first

“Pick 3 of 10 and A must be included”: place A, then choose 2 from the remaining 9, so the answer is C(9,2) = 36 – not C(10,3) minus 1.
“A must not be included” means choose 3 from the other 9. “Three people must stand together” uses the block method: treat them as one item, arrange, then multiply by 3!. “Three must not stand together” uses the gap method: arrange the others first, then insert into the gaps.Ask the student: why multiply by 3! after blocking? (The three inside the block can still swap.)

Student task: write the “A included” and “A excluded” cases side by side and check they add to C(10,3).

04 Question types and worked steps

Type	Example	Steps	Answer
Pure combination	choose 3 of 10	order irrelevant, use nCr	120 ways
Pure permutation	arrange 3 of 10	order matters, use nPr	720 ways
With a restriction	A must be included	place A then choose the rest	36 ways
Repeated items	arrangements of PASSION	divide by repeated factorials	7! / (2! x 2!)

Sanity check (teach this): A combination count must be a whole number and no larger than the matching permutation count: 120 <= 720. When unsure, check symmetry C(n,r) = C(n,n-r). In case-split questions the cases must add up to the unrestricted total; if they do not, a case is missing. "At least one" is safest by complement: total minus the count with none of the named letters.

05 Independent practice (3 questions)

Question	Answer	What to watch
Value of C(10,3)	120	nCr = nPr / 3!
Value of P(7,2)	42	7 x 6
Pick 3 of 10 with A included	36 ways	place A first

06 Common mistakes and what to say

Mistake	Student writes	Say this
Unordered treated as ordered	answering 720 for choosing representatives	“Choosing people does not care about order, so divide by 3!.”
Missing the divide for repeats	answering 6! for BANANA	“Three As are identical, without dividing you count the same word twice.”
Restriction handled by minus one	answering 119 for “A included”	“It is not one answer you subtract; place A first, then choose.”
A missing case	counting only the adjacent case	“Write both cases, the total is the sum.”

07 Homework (paper, section and question numbers)

Secondary maths chapter index — questions 3 and 5 are the same type
Prep guide home — other chapters in this subject
Primary maths test papers — warm up the multiplication principle

08 Exit check (last 5 minutes)

Question	Answer	Decision
Value of C(8,2)	28	all three correct means move to probability
Arrangements of 5 people in 5 seats	120	one wrong means reteach factorials
Strategy for “at least one”	total minus none	two wrong means back to act one

09 Tutor FAQ

When do I use nCr instead of nPr?

Ask whether order matters: use nPr when it does, nCr when it does not, or count ordered and divide by r!.

Why is “A must be included” not C(10,3) minus 1?

Because the excluded set is “A absent”, which is C(9,3) cases, not a single case.

Why divide by the factorial of the number of groups?

Swapping whole groups does not create a new grouping, so dividing by 3! avoids double counting.

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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.