Binomial expansion formula: (a + b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ
The binomial expansion formula is (a + b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ, with general term T(r+1) = nCr aⁿ⁻ʳ bʳ.
What each letter means
- \(n\) the power (a positive whole number)
- \(\binom{n}{r}\) the binomial coefficient nCr
- \(r\) which term: T₁ has r = 0
- \(a,\ b\) the two terms in the bracket, signs included
When to use it
To expand a bracket raised to a power without multiplying it out, or to find one term or coefficient (for example the x³ term) straight away.
Worked example
Find the coefficient of \(x^2\) in the expansion of \((1+3x)^5\).
- The \(x^2\) term has \(r=2\): \(T_3={}^5C_2\,(1)^3(3x)^2\)
- \({}^5C_2=10\) and \((3x)^2=9x^2\)
Answer: \(10\times9=90\)
Common mistake
Forgetting to raise the number with x to the power: in (1 + 3x)⁵ the r = 2 term has (3x)² = 9x², not 3x².
On your course
| Course | In the exam |
|---|---|
| Class 11 | Learn it: CBSE gives no formula booklet |
From our own CBSE Maths formula sheets, in our words.
Practise and revise
- Revise Class 11 Binomial Theorem
- Print Class 11 one-page formula sheet
Questions
What is the binomial expansion formula?
The binomial expansion formula is (a + b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ, with general term T(r+1) = nCr aⁿ⁻ʳ bʳ. n: the power (a positive whole number); nCr: the binomial coefficient nCr; r: which term: T₁ has r = 0; a, b: the two terms in the bracket, signs included.
Is the binomial expansion given in the exam?
Class 11: learn it: CBSE gives no formula booklet. This comes from our own CBSE Maths formula sheets; your teacher has the official booklet.
How do I find one term without expanding everything?
Use the general term T(r+1) = nCr aⁿ⁻ʳ bʳ and choose r so that the power of x is the one you want.
Our own wording, examples and card, checked by CBSE Math Revision. Not produced or endorsed by CBSE or NCERT.