HKDSE Maths: Binomial Theorem Lesson-Prep Guide (for tutors)

2026-11-01

Tutor lesson-prep guides · Secondary maths

Students can quote C(n,r) but lose the constant term once the bracket has x in the denominator. This lesson builds one habit: write the general term, make the index a function of r, then set it equal to the target power.

Objectives — Students can (1) write the general term of (a + b)^n as C(n,r) a^(n-r) b^r; (2) use it to find the coefficient of a stated power and the constant term; (3) find coefficient sums by substitution and decide the sign of each coefficient.

01 Prerequisites (3-minute check)

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

Skill	Quick check	Pass mark
Combinations	What is C(5,3)?	10
Index laws	Coefficient part of (2x)^3	8
Linear equations	Solve 6 – 2r = 0	r = 3

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-8 min	Pattern of the expansion	Pascal triangle and C(n,r)	write out (a+b)^3
8-20 min	The general term	the T(r+1) form	students write it once
20-34 min	Coefficient of a power	(1+2x)^5, x cubed	get 80
34-46 min	Constant term	(x+3/x)^6	get 540
46-56 min	Coefficient sums and signs	substitute 1 or -1	sum is 243
56-60 min	Exit check	three questions, marked at the door	independent

03 Teaching script (read it out)

Scene 1: From Pascal triangle to combinations

The coefficients of (a + b)^2 are 1, 2, 1 and of (a + b)^3 are 1, 3, 3, 1, which are exactly C(3,0) to C(3,3). So the coefficient of the (r+1)th term of (a + b)^n is C(n,r). That is the whole theorem; nothing extra needs memorising.

Ask the student: what is C(6,2), and why are the two middle coefficients of (a+b)^6 equal? (Symmetry: C(6,2) = C(6,4).)

Scene 2: The general term handles every question

General term: T(r+1) = C(n,r) a^(n-r) b^r with r starting at 0.
(1 + 2x)^5 has a = 1, b = 2x, n = 5, so T(r+1) = C(5,r) 2^r x^r. For the x cubed term, r = 3, giving C(5,3) x 2^3 = 10 x 8 = 80.
(2 – x)^7: coefficient of x to the fourth is C(7,4) x 2^3 x (-1)^4 = 35 x 8 = 280.

Warn: identify a and b explicitly, and carry the sign of b all the way through.

Scene 3: Constant terms and coefficient sums

(x + 3/x)^6: T(r+1) = C(6,r) x^(6-r) (3/x)^r = C(6,r) 3^r x^(6-2r). The constant term needs the index to be zero: 6 – 2r = 0, so r = 3 and the term is C(6,3) x 3^3 = 20 x 27 = 540.
For the sum of all coefficients, put x = 1: (1 + 2)^5 = 243. To get the difference between odd and even coefficients, put x = -1 and combine the two results.

Ask the student: constant term of (x + 2/x)^4? (r = 2, C(4,2) x 2^2 = 24.)

04 Question types and worked steps

Type	Example	Steps	Answer
Find a coefficient	stated power	general term, set the index, substitute r	80
Constant term	positive and negative powers mixed	set the index to zero and solve for r	540
Coefficient sum	all coefficients added	substitute x = 1	243
Existence	does a certain term exist	r must be a whole number from 0 to n	r = 3 exists

Sanity check (teach this): The r you solve must be a whole number between 0 and n, otherwise the term does not exist and the answer is that there is no such term. A constant term needs a positive power to cancel a negative one, and a coefficient sum must be the same order of magnitude as (a + b)^n.

05 Independent practice (3 questions)

Question	Answer	What to watch
coefficient of x cubed in (1 + 2x)^5	80	r = 3, C(5,3) x 8
constant term of (x + 3/x)^6	540	6 – 2r = 0, so r = 3
sum of all coefficients of (1 + 2x)^5	243	substitute x = 1

06 Common mistakes and what to say

Mistake	Student writes	Say this
index and r swapped	uses C(5,2)	In T(r+1) the r belongs to b, so keep the powers straight.
forgetting to raise the coefficient	answers 20 instead of 80	(2x)^3 is 8x cubed, not 2x cubed.
setting r = 0 for the constant term	answers C(6,0)	The index must be zero, not r.
dropping a minus sign	answers plus 280	A negative b makes every odd-power term negative.

07 Homework (paper, section and question numbers)

DSE maths practice paper (binomial theorem) — Paper 1 questions 9 and 10 (coefficients and constant terms)
Secondary maths chapter index — all chapters

08 Exit check (last 5 minutes)

Question	Answer	Decision
coefficient of x cubed in (1 + 2x)^5	80	all three right, move to probability
constant term of (x + 3/x)^6	540	one wrong, re-teach the general term
value to substitute for the coefficient sum	x = 1	two wrong, restart act one

09 Tutor FAQ

Must students memorise the Pascal triangle?

Not deeply. The coefficients are C(n,r); the triangle only builds intuition.

What if n is large?

Use the nCr key on the calculator, which is allowed; the marks are for the general term.

Can there be two constant terms?

No. In a single binomial expansion only one r makes the index zero.

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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.