Binomial theorem: general term and coefficients: Class 11 practice

What you should know first

• Combinations
• Laws of exponents

The idea behind binomial theorem: general term and coefficients

(a + b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ. The (r + 1)th term is Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ — use it to find any single term or coefficient without full expansion.

Step-by-step method

1. Write Tᵣ₊₁ = ⁿCᵣ (first)ⁿ⁻ʳ (second)ʳ.
2. Collect the power of x and set it equal to the required power.
3. Solve for r and evaluate.

Worked example 1 (Basic)

Find the coefficient of x^1 in (2x + 1)^4.

1. Term with x^1: 4C3 (2x)^1 (1)^3
2. = 4 × 2 × 1
3. Coefficient = 8
4. Answer: 8

Worked example 2 (Standard)

Find the coefficient of x^5 in (x + 1)^7.

1. Term with x^5: 7C2 (1x)^5 (1)^2
2. = 21 × 1 × 1
3. Coefficient = 21
4. Answer: 21

Worked example 3 (Challenge)

Find the coefficient of x^2 in (x − 3)^4.

1. Term with x^2: 4C2 (1x)^2 (-3)^2
2. = 6 × 1 × 9
3. Coefficient = 54
4. Answer: 54

Common mistakes and misconceptions

• Calling Tᵣ the rth term with ⁿCᵣ (it is the (r+1)th)
• Forgetting the sign of a negative second term

Practice questions

1. (Basic) Find the coefficient of x^4 in (3x + 3)^7.
2. (Basic) Find the coefficient of x^4 in (3x + 1)^5.
3. (Standard) Find the coefficient of x^5 in (3x + 1)^7.
4. (Standard) Find the coefficient of x^2 in (3x + 1)^8.
5. (Challenge) Find the coefficient of x^1 in (2x − 3)^4.
6. (Challenge) Find the coefficient of x^4 in (3x − 2)^7.

Answer key with full working

• Q1: Term with x^4: 7C3 (3x)^4 (3)^3 → = 35 × 81 × 27 → Coefficient = 76545 → Answer: 76545
• Q2: Term with x^4: 5C1 (3x)^4 (1)^1 → = 5 × 81 × 1 → Coefficient = 405 → Answer: 405
• Q3: Term with x^5: 7C2 (3x)^5 (1)^2 → = 21 × 243 × 1 → Coefficient = 5103 → Answer: 5103
• Q4: Term with x^2: 8C6 (3x)^2 (1)^6 → = 28 × 9 × 1 → Coefficient = 252 → Answer: 252
• Q5: Term with x^1: 4C3 (2x)^1 (-3)^3 → = 4 × 2 × -27 → Coefficient = -216 → Answer: -216
• Q6: Term with x^4: 7C3 (3x)^4 (-2)^3 → = 35 × 81 × -8 → Coefficient = -22680 → Answer: -22680

Syllabus note

This skill is listed under "Binomial Theorem" for Class 11 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.

Related practice and notes

• Binomial Theorem chapter notes (Class 11)
• Middle term and term independent of x
• Expanding a binomial fully
• Permutations and combinations problems
• Counting with Venn diagrams
• Domain and range of functions
• Radians, degrees and arc length
• All Class 11 practice

Frequently asked questions

Which chapter is binomial theorem: general term and coefficients in?

"Binomial Theorem", Class 11 Maths (NCERT/CBSE). Other boards may place it differently.

How were the answers checked?

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.