Q1. Find the 3rd term in the expansion of (x + 2)⁵.
3rd term means T(r+1) with r = 2, so r = 2, n = 5
T₃ = ⁵C₂ · x⁵⁻² · 2² = 10 · x³ · 4
T₃ = 40x³
Binomial Theorem — Class 11
Expanding (a+b)¹⁰ by hand is painful — the binomial theorem gives any term instantly. Compute any term of any expansion here.
Binomial theorem(a+b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ for r = 0 to n — a sum of n+1 terms.
General termT(r+1) = ⁿCᵣ aⁿ⁻ʳ bʳ — the (r+1)th term of the expansion (note it starts counting at r = 0).
Pascal's triangleEach row gives the coefficients ⁿC₀, ⁿC₁, …, ⁿCₙ for that value of n.
General-term (coefficient) calculator Interactive
For (a+b)ⁿ, find the (r+1)th term's binomial coefficient ⁿCᵣ.
Coefficient, ⁿCᵣ—
r = 0 gives the FIRST term, r = 1 the second, and so on — the general term is T(r+1), so it is one ahead of r.
Solved examples
Formula sheet
| Quantity | Formula |
|---|---|
| Expansion | (a+b)ⁿ = Σᵣ ⁿCᵣ aⁿ⁻ʳ bʳ |
| General term | T(r+1) = ⁿCᵣ aⁿ⁻ʳ bʳ |
| Number of terms | n + 1 |
Common mistakes & exam wins
- The expansion of (a+b)ⁿ has exactly n + 1 terms.
- To find the "kth term", use r = k − 1 in the general term formula T(r+1).
- The coefficients of (a+b)ⁿ are exactly the nth row of Pascal's triangle.
Frequently asked questions
What is the binomial theorem?
A formula for expanding (a+b)ⁿ as a sum: Σ ⁿCᵣ aⁿ⁻ʳ bʳ for r from 0 to n.
What is the general term in a binomial expansion?
T(r+1) = ⁿCᵣ aⁿ⁻ʳ bʳ, the (r+1)th term of the expansion.
How many terms are in the expansion of (a+b)ⁿ?
Exactly n + 1 terms.