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IGCSE Add Maths 0606

Common Mistakes in IGCSE Add Maths Binomial Expansion

Five specific errors students repeat in binomial expansion questions, and the exact step where each one happens.

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From the lessons of Teacher Au, teaching Cambridge IGCSE Maths since 2018 · Updated 6 November 2026

Short answer

What are the most common mistakes in IGCSE Add Maths binomial expansion?

The five most common binomial expansion mistakes in IGCSE Additional Mathematics (0606) are: using the wrong value of r when finding a specific term, sign errors with negative terms inside the bracket, forgetting to raise the coefficient to the correct power, confusing nCr with n × r, and failing to identify the term independent of x. Each is a process error that appears in the middle of the formula, not at the formula recall step.

Source: Cambridge IGCSE Additional Mathematics (0606) syllabus for 2025 to 2027 (PDF).

Mark analysis

Why binomial expansion is a reliable mark-loss topic in IGCSE Add Maths

Binomial expansion is part of the series topic (topic 12) of the Cambridge IGCSE Additional Mathematics (0606) syllabus, and the general term is given in the exam’s list of formulas. Students who know the formula but cannot execute the steps consistently lose marks not because they do not understand binomial expansion, but because they rush through the r substitution, miss a negative sign, or forget that the coefficient inside the bracket is raised to a power, not multiplied by it.

Mistake 1

Using the wrong value of r when finding a specific term

The general term in the binomial expansion of (a + b)n is Tr+1 = nCr an−rbr. The term number is r + 1, not r. Students who confuse the term number with r consistently find the wrong term.

  • Where it goes wrong. A question asks for the 4th term. Students set r = 4, which gives T5, the 5th term, when they should set r = 3 to find T3+1 = T4. This shifts every later calculation by one position.
  • The fix. Write out the general term label Tr+1 before substituting. If the question asks for the 4th term, write “4th term: Tr+1 = T4, so r = 3” before using the formula. The extra line prevents the off-by-one error that can cost the whole question.
  • Exam pattern. Questions that ask for a specific term (3rd, 4th, 5th) are a common binomial question type.
Mistake 2

Sign error when the b term is negative

When the bracket is of the form (a − b), the general term must be written for (a + (−b)). Students who write the general term without tracking the sign of b get incorrect signs on alternate terms of the expansion.

  • Where it goes wrong. In the expansion of (2x − 3)5, the b term is −3. A student who substitutes b = 3 (dropping the negative) gets a positive coefficient where the correct answer is negative, on every term with an odd power of b.
  • The fix. Write b explicitly with its sign in the general term before substituting. For (2x − 3)5, write b = −3, then substitute. The sign is part of b, not a separate step to remember at the end.
  • Pattern across terms. The sign alternates predictably in an (a − b)n expansion: positive, negative, positive, negative. If the student's expansion does not alternate, the sign of b was almost certainly dropped.
Mistake 3

Forgetting to raise the coefficient inside the bracket to the correct power

When the bracket contains a term like (3x), the binomial formula needs the entire term 3x inside the power, as (3x)r or (3x)n−r, not just the x. Students who split the coefficient from the variable lose the power on the numerical part.

  • Where it goes wrong. In (3x + 2)4, the a term is 3x. When r = 2, the formula gives (3x)2 = 9x2. Students who write x2 (dropping the 32 = 9) get the coefficient wrong by a factor of 9.
  • The fix. Bracket the entire a or b term before raising it to a power. Write (3x)2, not 3x2. The bracket reminds the student that both the 3 and the x are being squared.
  • Compounding error. When both terms carry a number, as in (3x + 2)4, the error can appear twice in the same term: once in (3x)n−r and once in 2r.
Mistake 4

Confusing nCr with n × r

nCr (read as “n choose r”) is a combinatorial coefficient calculated as n! ÷ (r!(n − r)!). It is not the same as n × r. Students who have not used nCr enough to build fluency sometimes substitute incorrectly.

  • Where it goes wrong. For n = 6, r = 2: 6C2 = 15. Students who compute 6 × 2 = 12 get a wrong coefficient on that term. This error is easier to make on Paper 1, where no calculator is allowed, or when the student has not practised Pascal's triangle as a cross-check.
  • The fix. Know Pascal's triangle up to n = 6 by heart. Use it as a cross-check for small values of n. For larger n, write out the factorial formula explicitly before simplifying. Never compute nCr mentally without checking it against the triangle or the formula.
  • Related confusion. Students sometimes confuse 8C3 with 8C5 and get the same answer (they are equal by symmetry: nCr = nCn−r), which is fine. The error is computing nCr as a product rather than a quotient.
Mistake 5

Failing to find the term independent of x

Questions that ask for the "term independent of x" (or "constant term") require the student to find the value of r that makes the power of x equal to zero. Students who do not set up the x power equation correctly find the wrong term or the wrong coefficient.

From our lessons

1Binomial expansionAdd Maths 0606

Why does my child pick the wrong term for “the term independent of x”?

The question

Find the term independent of x in the expansion of (x + 2 over x)6.

Your child’s working

  1. x6−r: 6 − r = 0marked wrong
  2. r=6
  3. term=26 = 64

Teacher’s note: 2 over x has an x in it too

The marked fix

  1. x6−r × (x−1)r=x6−2r
  2. 6 − 2r=0, so r = 3
  3. term=6C3 × 23
  4. =20 × 8 = 160correct
Why it happens
The child counts the power of x in the first term of the bracket only. The second term, 2 over x, also carries x, so its power has to be counted too.
How Teacher Au fixes it in class
He tells students they do not need to expand everything, only find the one term they want. They write the power of x from both parts of the bracket as one equation and solve it for r.
A 2-minute check at home
Ask your child to put their r back into the general term and check that the x really cancels out. With r = 6 the term is 64 over x to the power 6, which still has x in it.
  • Where it goes wrong. In the expansion of (x + 3 over x squared)6, the power of x in the general term is (6 − r) − 2r = 6 − 3r. Setting this equal to zero gives r = 2, so the term independent of x is 6C2 × 32 = 135. Students who set up the power of x incorrectly (for example, treating the x2 in the denominator as a positive power) get 6 + r = 0, which has no valid r, and then guess a term.
  • The fix. Write out the power of x as a separate equation before substituting. For each term, label: power of x = (contribution from a term) + (contribution from b term). Set equal to zero and solve for r. This equation takes four seconds to write and removes a common error in "term independent of x" questions.
  • Exam pattern. The syllabus itself uses “find the term independent of x” as its example of the general term. It is a predictable question type that students should practise as a template, not as a novel problem each time.
What this means for your child

These are process errors, not concept gaps

Every mistake above is about skipping or misapplying a step in a written formula, not about failing to understand what binomial expansion is. Students who make these errors typically know the general term formula. The problem is that they rush through the substitution, treating familiar-looking steps as safe to do mentally.

In class, Teacher Au addresses this by requiring students to write every labelled step of the general term before any simplification: Tr+1, the explicit value of r, the a and b terms with signs and brackets, and the nCr calculation. The point is not slowness. It is that every mark lost to a binomial error is a process error that would not have happened if the step had been written down.

The goal is exam consistency: a student who can carry out the binomial template reliably under time pressure stops losing marks to these steps.

To see the same marking across every topic, look through all the maths mistakes we log, by topic.

Related: how to pass IGCSE Additional Mathematics, how to revise for the Add Maths exam, and the common mistakes in permutations and combinations and vectors.

Related reading

What is the hardest topic in IGCSE Add Maths?

Binomial expansion is not the hardest topic, but it is one of the most reliable mark-loss topics because the errors are mechanical. See the full difficulty breakdown by topic.

Read the topic difficulty guide
Related reading

Why students lose marks in IGCSE Add Maths

The same pattern of process errors in the middle steps shows up across topics, not just binomial expansion. See how it happens in algebra and calculus.

Read the mark-loss analysis
Questions parents ask

Frequently asked questions

The five most common mistakes are: using the wrong r value for a specific term, sign errors with negative brackets, forgetting to raise the coefficient inside the bracket to the correct power, confusing nCr with n × r, and failing to correctly identify the term independent of x. Each is a process error in the middle of the formula, not a concept gap.

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