Binomial Theorem Class 11 Notes: General Term, Middle Term and Tricks

Binomial Theorem is a 4–6 mark chapter where every question is ‘find the term with property X’. Learn the general term and the rest follows.

The theorem

(a + b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ, r = 0 to n; n + 1 terms. General term T_r+1 = nCr aⁿ⁻ʳ bʳ.

Standard results

• Sum of coefficients: put x = 1 → 2ⁿ. Sum of odd-position coefficients = sum of even = 2ⁿ⁻¹.
• Middle term: n even → T_(n/2)+1; n odd → two middle terms T_(n+1)/2 and T_(n+3)/2.
• nCr = nC(n−r); coefficients symmetric.
• (1 + x)ⁿ ≈ 1 + nx for small x (approximation).

Solved examples

1. Term independent of x in (x² + 1/x)⁹: T_r+1 = 9Cr x^(18−2r) x^(−r) = 9Cr x^(18−3r); 18 − 3r = 0 → r = 6 → 9C6 = 84.

2. Coefficient of x⁵ in (1 + 2x)⁸: r = 5 → 8C5 × 2⁵ = 56 × 32 = 1792.

3. Middle term of (x/2 + 2/x)¹⁰: T₆ = 10C5 (x/2)⁵(2/x)⁵ = 252.

4. Show 9ⁿ⁺¹ − 8n − 9 is divisible by 64: 9ⁿ⁺¹ = (1 + 8)ⁿ⁺¹ = 1 + 8(n + 1) + 64(…) → 9ⁿ⁺¹ − 8n − 9 = 64(…). ✓

5. (1.01)¹⁰⁰⁰⁰⁰ vs 10000: (1 + 0.01)¹⁰⁰⁰⁰⁰ > 1 + 100000 × 0.01 = 1001 … and in fact vastly larger — use the first two terms to compare.

6. Greatest term in (1 + x)ⁿ: find r with (n − r + 1)x/r ≥ 1.

Tips

• Write the general term first, always.
• Simplify the power of x to a single exponent before setting a condition.
• For ‘coefficient of xᵏ’ in a product, expand only the needed terms.

Common mistakes

• Using T_r instead of T_r+1
• Forgetting the coefficient inside the bracket (2x, not x)
• Choosing only one middle term for odd n

Related: Sequences and series.

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Frequently asked questions

Are these Binomial Theorem Class 11 Notes notes enough for the exam?

They cover every formula, definition and question type that appears in school and board papers for this chapter. Use them with the NCERT examples, exercise questions and the last five years' papers to be fully prepared.

How should I revise Binomial Theorem Class 11 Notes?

Revise the notes after 1 day, 7 days and 30 days. Solve five questions each time instead of only reading, and keep an error list of the steps you get wrong.

What if I get stuck on a binomial theorem question?

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