How is the general term written?

The term with r + 1 in (a + b)ⁿ is n choose r times a to the n minus r times b to the r.

How do I find a constant term?

Set the power of x in the general term to zero, solve for r, then substitute back.

S6 Binomial Theorem, Paper 1: general term and coefficients

2026-11-02

Tutor Prep Guide · Primary English

The general term is n choose r times a to the n minus r times b to the r, so write it down before matching powers.

Focus　Never drop the power on b, and the constant term is the one where the power of x is zero.

01　Paper details (check answers afterwards)

Item	Detail
Questions	10
Suggested time	25
Difficulty mix	Foundation 4 · Intermediate 3 · Challenge 2 · Error redo 1
Chapter	Chapter: S6 Maths: Binomial Theorem
Level	S6

02　Questions (10 items)

FoundationExpand (a + b)³.
FoundationFind the coefficient of x² in (1 + x)⁵.
FoundationFind the coefficient of x³ in (1 + 2x)⁴.
FoundationFind the fourth term in the expansion of (x + 1)⁶.
IntermediateFind the coefficient of x² in (1 + x)⁸.
IntermediateFind the coefficient of x³ in (2 + x)⁵.
IntermediateFind the constant term in the expansion of (x + 2/x)⁴.
ChallengeFind the coefficient of x⁵ in (1 + x)³(1 + x)⁴.
ChallengeIf the coefficient of x² in (1 + x)ⁿ is 45, find n.
Error redoError: a student gives 30 as the coefficient of x² in (1 + 3x)⁵. What is correct?

03　Answers and one-line notes

#	Answer	One-line note
1	a³ + 3a²b + 3ab² + b³	The coefficients run 1, 3, 3, 1.
2	10	Take 5 choose 2, which is 10.
3	32	Four choose three times 2 cubed, which is 4 times 8.
4	20x³	The fourth term takes 6 choose 3 times x cubed, giving 20x³.
5	28	Take 8 choose 2, which is 28.
6	40	Five choose three times 2 squared, which is 10 times 4.
7	24	The general term is 4 choose r times 2 to the r times x to the 4 minus 2r, and r equal to 2 removes x.
8	21	Combine first into (1 + x)⁷, then take 7 choose 5, which is 21.
9	n = 10	Set n(n minus 1) over 2 equal to 45, so n squared minus n minus 90 is zero and n is 10.
10	90	Multiply by 3 squared rather than 3, so 10 times 9 gives 90.

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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