Binomial Theorem
Class 11 Maths • Chapter 07 • Comprehensive Interactive Notes
1. Binomial Theorem
The Binomial Theorem gives a formula for expanding \( (a+b)^n \) for any positive integer \( n \).
\( (a+b)^n = \sum_{r=0}^{n} {^nC_r} a^{n-r} b^r \)
Where \( ^nC_r \) are binomial coefficients.
Expansion Engine
Expand \( (a+b)^n \).
2. Pascal's Triangle
The coefficients of the expansion form a pattern known as Pascal's Triangle.
3. General & Middle Terms
| Term Type |
Formula / Condition |
| General Term |
\( T_{r+1} = {^nC_r} a^{n-r} b^r \) |
| Middle Term (n even) |
\( (\frac{n}{2} + 1)^{th} \) term |
| Middle Term (n odd) |
\( (\frac{n+1}{2})^{th} \) and \( (\frac{n+1}{2} + 1)^{th} \) terms
|
4. Properties of Coefficients
\( ^nC_0 + ^nC_1 + ... + ^nC_n = 2^n \)
\( ^nC_0 + ^nC_2 + ... = ^nC_1 + ^nC_3 + ... = 2^{n-1} \)
⚠️ Common Mistakes to Avoid
- ✘ Forgetting that total terms = n + 1
- ✘ Wrong power of a or b in general term
- ✘ Confusing middle term for even & odd n
- ✘ Using wrong value of r (remember: term is r+1)
- ✘ Ignoring coefficient while finding numerical value
📌 NCERT Exam Question Types
- ✔ Find a particular term independent of x
- ✔ Find coefficient of a given power
- ✔ Find middle term(s)
- ✔ Prove identities using binomial expansion
- ✔ Numerical evaluation using \( (1+x)^n \)
CBSE Insight:
Questions are rarely direct expansions.
CBSE tests term identification, coefficient logic, and simplification.
Concept Mastery Quiz
1. The total number of terms in \( (a+b)^n \) is:
2. The middle term in \( (a+b)^6 \) is:
3. The general term is denoted by:
4. \( ^nC_0 + ^nC_n \) equals:
5. The coefficient of the middle term is:
🧠 One-Page Revision Checklist
- ✔ Expansion formula remembered
- ✔ General term formula clear
- ✔ Middle term logic (even / odd n)
- ✔ Pascal triangle relation understood
- ✔ Coefficient properties revised
- ✔ Able to solve term-based questions
Self-Test:
- ❓ How many terms in \( (a+b)^9 \)?
- ❓ Middle term in \( (x+y)^8 \)?
- ❓ Coefficient of \( x^3 \) in \( (1+x)^5 \)?