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Binomial Theorem

The general term does half the work, coefficient sums the other half. Nearly half its questions have numeric answers, so accuracy matters more than speed here.

Questions in the bank
165
q/paper in 2025–26
1.05
Numeric answer
45%
Notes pages
8

Tier: Core

When you’ll see it

A power of a bracket: one coefficient, the constant term, a sum of C(n, r) terms, or the remainder of a large power.

How this chapter is tested

The general term T(r + 1) = C(n, r) aⁿ⁻ʳ bʳ does most of the chapter's work. Set the power of x to what is asked, solve for r, and read the coefficient. Consecutive coefficients, products of brackets and rational terms are the same step with a different condition on r.

The second half is sums of binomial coefficients. Substitute x = 1, −1 or a complex root into an expansion; differentiate it for r·C(n, r); integrate it for C(n, r)/(r + 1); multiply two expansions for a product of coefficients. Once you name which of these a sum is, it closes in two lines. The time goes on naming it.

Most answers are whole numbers, so the chapter suits numeric-answer questions. Remainders of large powers write the base as a multiple of the divisor plus or minus 1 and expand. The coefficient sums share their tools with Sequences and Series and with Definite Integration.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • The general term

    T(r + 1) = C(n, r) aⁿ⁻ʳ bʳ; set the power of x and solve for r. Simplify the bracket first when it factors.

  • Consecutive coefficients and special terms

    C(n, r)/C(n, r − 1) = (n − r + 1)/r turns three consecutive coefficients into two linear equations; middle and end terms by symmetry.

  • Products and three-term brackets

    A coefficient of a product is a short sum of products; a three-term bracket is factored or expanded with the multinomial term.

  • Rational terms and integral parts

    A term is rational when both exponents are whole; (a + √b)ⁿ + (a − √b)ⁿ is an integer.

  • Coefficient sums

    x = 1 gives the sum of all coefficients, x = −1 the alternating sum; differentiate for Σ r·C(n, r) = n·2ⁿ⁻¹.

  • Fractions and products of coefficients

    Integrate for C(n, r)/(r + 1); Σ C(m, r) C(n, k − r) = C(m + n, k).

  • Hockey stick and geometric sums

    C(r, r) + C(r + 1, r) + … + C(n, r) = C(n + 1, r + 1); a sum of powers of (1 + x) is a geometric series.

  • Remainders and divisibility

    Write the base as kd ± 1, expand, and only the last term survives division by d.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • The index is r + 1

    The term containing bʳ is the (r + 1)-th. 'The 7th term' means r = 6.

  • The sign of the second term

    In (x − 1/x)ⁿ each term carries (−1)ʳ. Dropping it flips the sign of every odd-r coefficient.

  • Only one exponent checked

    For a rational term in (a^(1/p) + b^(1/q))ⁿ, both (n − r)/p and r/q must be whole numbers.

  • A left-over factor

    2¹⁰⁰ = 2 · 8³³, so its remainder on division by 7 is 2, not 1. A factor outside the bracket stays in the answer.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Binomial Theorem notes

Drill every Binomial Theorem question

165 questions from the bank, across 8 subtopics.

Drill the 165 questions

Drill one subtopic at a time

The 8 subtopics, in teaching order.

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