Binomial Theorem

Concept overview

The binomial theorem states that (a + b)ⁿ = Σ C(n, k) a^(n−k) b^k for k = 0 to n. Each term has a binomial coefficient C(n, k), which counts the number of ways to choose k of the n multiplication slots to contribute b.

The general term (r + 1th term) is T_{r+1} = C(n, r) a^(n−r) b^r. Setting the power condition gives the coefficient of a specific term. The middle term is T_{(n/2) + 1} when n is even, or two middle terms T_{(n+1)/2} and T_{(n+3)/2} when n is odd.

Sum of coefficients in (a + b)ⁿ is obtained by putting a = b = 1: gives 2ⁿ. Sum of even-indexed minus odd-indexed coefficients: put a = 1, b = −1: gives 0 (for n ≥ 1). These identities generate many JEE questions.

The theorem extends to non-integer n as an infinite series (Newton's binomial series) provided |b/a| < 1, but this is used less in JEE Main and only marginally in Advanced.

Key formulas & concepts

• Binomial expansion: (a + b)ⁿ = Σ C(n, k) a^(n−k) b^k
• General term: T_{r+1} = C(n, r) a^(n−r) b^r
• Middle term (n even): T_{(n/2) + 1}
• Sum of all coefficients: = 2ⁿ (set a = b = 1)
• Alternating sum: Σ (−1)^k C(n, k) = 0 for n ≥ 1
• Coefficient of a specific power: match the exponents in T_{r+1}

Common mistakes to avoid

• Getting the general term index off by one — T_{r+1} corresponds to power r of b (not r of a).
• Forgetting the sign of C(n, k)(−1)^k when the expansion is (a − b)ⁿ.
• Applying the infinite-series form (Newton's binomial) outside its convergence range.
• Confusing coefficient of x^k with the k-th term — the general-term index doesn't equal the exponent.

JEE Main relevance

JEE Main asks 1–2 binomial problems every paper — usually one general-term/middle-term question and one specific-coefficient calculation.

JEE Advanced relevance

Advanced problems combine binomial coefficients with combinatorial identities (Vandermonde's identity, etc.) or ask about sums of products of coefficients.

Previous-year questions (PYQ)

Filter the PYQ bank by 'Binomial Theorem'. Coefficient-of-a-specific-power problems are the highest-recurrence archetype.