Chapter 7 of 14

50%

CBSE Class 11 Mathematics★ 6-8 marks · CBSE Class 11 Mathematics Chapter 7

Binomial Theorem

(a + b)ⁿ — how do you expand it WITHOUT multiplying a billion times? The Binomial Theorem gives the answer: use Pascal's triangle and nCr coefficients. CBSE Class 11 Mathematics Chapter 7.

⏱ 20 min readMedium difficulty✓ Free · NCERT-aligned

Ask AI about this

🔤Build your vocabulary from this chapterLexiGenome mines the key words, explains them in your language, and schedules them with spaced repetition.Mine words →

By the end of this chapter you'll be able to…

1Construct Pascal's Triangle and state its relationship to binomial coefficients nCr
2State and apply the Binomial Theorem to expand (a+b)ⁿ for positive integer n
3Use the general term formula Tᵣ₊₁ = nCᵣ·aⁿ⁻ʳ·bʳ to find any specified term in an expansion
4Identify the middle term(s) of a binomial expansion for both even and odd n
5Apply special cases (1+x)ⁿ and (1−x)ⁿ and use the sum-of-coefficients = 2ⁿ property

💡

Why this chapter matters

The Binomial Theorem provides a systematic formula for expanding (a+b)ⁿ for any positive integer n, avoiding repeated multiplication. The general term formula is the key tool for finding any specific term in an expansion — a type of question that appears in virtually every JEE Main paper.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

🔗

Permutations and Combinations

Binomial coefficients nCr are the combination values; Pascal's identity nCr+nC(r−1)=(n+1)Cr underpins the theorem

Binomial Theorem

"Algebra begins with (a + b)². The Binomial Theorem is what happens when the 2 becomes n."

1. Chapter Overview

Expanding (a + b)² = a² + 2ab + b² is easy. But what about (a + b)¹⁰? (a + b)ⁿ? The Binomial Theorem provides the formula: the expansion uses BINOMIAL COEFFICIENTS (nCr) and PATTERNS in the powers. This chapter covers: the theorem, Pascal's triangle, general and middle terms.

2. Pascal's Triangle

A triangular array where each number is the SUM of the two numbers above it:

        1          ← (a + b)⁰
       1 1         ← (a + b)¹
      1 2 1        ← (a + b)²
     1 3 3 1       ← (a + b)³
    1 4 6 4 1      ← (a + b)⁴
   1 5 10 10 5 1   ← (a + b)⁵

Row n (starting from 0) gives the coefficients of (a + b)ⁿ
Each number = nCr. Row 4: ⁴C₀=1, ⁴C₁=4, ⁴C₂=6, ⁴C₃=4, ⁴C₄=1

3. The Binomial Theorem

For any positive integer n:

$(a + b)^{n} = \sum_{r = 0}^{n} (r n) a^{n - r} b^{r}$

Expanded: $(a + b)^{n} =^{n} C_{0} a^{n} +^{n} C_{1} a^{n - 1} b +^{n} C_{2} a^{n - 2} b^{2} + ... +^{n} C_{n} b^{n}$

Observations

There are (n + 1) terms in the expansion
In each term: power of a DECREASES from n to 0. Power of b INCREASES from 0 to n.
Sum of powers of a and b in each term = n
The coefficients are SYMMETRIC: nCr = nC(n-r)
nC₀ = nCn = 1

4. General Term

The (r + 1)th term (counting from r = 0): $T_{r + 1} =^{n} C_{r} \cdot a^{n - r} \cdot b^{r}$

Middle Term(s)

If n is EVEN: there is ONE middle term. Term number = (n/2 + 1). r = n/2.
If n is ODD: there are TWO middle terms. Terms at r = (n-1)/2 and r = (n+1)/2.

5. Special Cases

(1 + x)ⁿ

$(1 + x)^{n} =^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} + ... +^{n} C_{n} x^{n}$

(1 — x)ⁿ

Same expansion but with ALTERNATING SIGNS: $(1 - x)^{n} =^{n} C_{0} -^{n} C_{1} x +^{n} C_{2} x^{2} - ... + (- 1)^{n}^{n} C_{n} x^{n}$

6. Properties of Binomial Coefficients

Sum of all coefficients: nC₀ + nC₁ + nC₂ + ... + nCn = 2ⁿ
Sum of coefficients at EVEN positions = sum at ODD positions = 2ⁿ⁻¹
nCr + nC(r-1) = (n+1)Cr

7. Exam Focus

Pascal's triangle — how constructed, relationship to nCr
Binomial theorem formula — expansion of (a + b)ⁿ
General term Tr+1 = nCr · a^(n-r) · b^r
Middle term(s) — formula for even and odd n
Expansion of (1 + x)ⁿ and (1 — x)ⁿ

8. Key Formulas

(a + b)ⁿ = Σ nCr a^(n-r) b^r (r = 0 to n)
General term: Tr+1 = nCr a^(n-r) b^r
Sum of coefficients = 2ⁿ

9. Conclusion

The Binomial Theorem is a PATTERN-MACHINE:

PASCAL'S TRIANGLE: The coefficients for small n — visible at a glance
THE THEOREM: For ANY n — a systematic expansion using nCr
GENERAL TERM: Find ANY term without writing the whole expansion

'The binomial theorem is a thing of beauty — a simple pattern that generates an infinity of terms from two numbers and a power.'

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Binomial Theorem

(a+b)ⁿ = Σ(r=0 to n) nCr · aⁿ⁻ʳ · bʳ = nC₀aⁿ + nC₁aⁿ⁻¹b + nC₂aⁿ⁻²b² + ... + nCₙbⁿ

(n+1) terms total; powers of a decrease, powers of b increase, sum of powers in each term = n

General Term (r+1)th term

Tᵣ₊₁ = nCᵣ · aⁿ⁻ʳ · bʳ

r starts at 0, so the 1st term is T₁ (r=0), the 2nd is T₂ (r=1), etc.

Middle Term — n even

If n is even: one middle term = T(n/2+1) at r = n/2

Expansion of (a+b)ⁿ has n+1 terms; the (n/2+1)th is the single middle term

Middle Terms — n odd

If n is odd: two middle terms = T((n+1)/2) at r=(n−1)/2 and T((n+3)/2) at r=(n+1)/2

For odd n, n+1 is even so there are exactly two central terms

Sum of Binomial Coefficients

nC₀ + nC₁ + nC₂ + ... + nCₙ = 2ⁿ

Obtained by substituting a=1, b=1 in (a+b)ⁿ

Pascal's Identity

nCᵣ + nCᵣ₋₁ = (n+1)Cᵣ

Each entry in Pascal's Triangle = sum of the two entries directly above it

⚠️

Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Starting the general term count from r=1 instead of r=0

✓ The general term is Tᵣ₊₁. At r=0, T₁ is the FIRST term. At r=1, T₂ is the SECOND term. Always start r at 0.

WATCH OUT

✗ Substituting the wrong exponents in Tᵣ₊₁ = nCᵣ·aⁿ⁻ʳ·bʳ

✓ In (a+b)ⁿ: power of a = n−r, power of b = r. They must always sum to n. Double-check: (n−r)+r=n.

WATCH OUT

✗ Forgetting to account for coefficients of a and b when they are not 1

✓ In (2x + 3y)⁴, the 'a' is 2x and the 'b' is 3y. So T₃ = ⁴C₂·(2x)²·(3y)² — raise the ENTIRE term (coefficient and variable together) to the power.

WATCH OUT

✗ Saying there is one middle term for any n

✓ For even n: ONE middle term. For odd n: TWO middle terms. The expansion has n+1 terms — if n+1 is odd (n even), there is one middle term; if n+1 is even (n odd), there are two.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Ex 7.1

Exercise 7.1

Expanding binomial expressions using the Binomial Theorem

Questions

Ex 7.2

Exercise 7.2

Finding general term, middle term, specific terms in the expansion

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· Expansion

Expand (x + 2)⁴ using the Binomial Theorem.

▶ Show solution

(x+2)⁴ = ⁴C₀x⁴ + ⁴C₁x³(2) + ⁴C₂x²(2²) + ⁴C₃x(2³) + ⁴C₄(2⁴) = x⁴ + 4×2x³ + 6×4x² + 4×8x + 16 = x⁴ + 8x³ + 24x² + 32x + 16.

Q2MEDIUM· General Term

Find the 5th term in the expansion of (2x − 3y)⁸.

▶ Show solution

T₅ = T(4+1): r=4. T₅ = ⁸C₄·(2x)⁸⁻⁴·(−3y)⁴ = 70·(2x)⁴·(−3y)⁴ = 70·16x⁴·81y⁴ = 70×16×81×x⁴y⁴ = 90720x⁴y⁴.

Q3HARD· Middle Term

Find the middle term(s) in the expansion of (x/2 + 3/x)⁸.

▶ Show solution

n=8 (even), so ONE middle term: T₅ (at r=4). T₅ = ⁸C₄·(x/2)⁴·(3/x)⁴ = 70·(x⁴/16)·(81/x⁴) = 70·81/16 = 5670/16 = 2835/8.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•(a+b)ⁿ has exactly (n+1) terms in its expansion
•General term: Tᵣ₊₁ = nCᵣ·aⁿ⁻ʳ·bʳ where r = 0, 1, 2, ..., n
•Coefficients are symmetric: nCᵣ = nCₙ₋ᵣ, so first and last coefficients are equal (=1)
•Sum of all binomial coefficients = 2ⁿ (put a=b=1)
•Pascal's Identity: nCᵣ + nCᵣ₋₁ = (n+1)Cᵣ — each Pascal's Triangle entry = sum of two above
•Middle term: one middle term if n is even (T(n/2+1)); two middle terms if n is odd
•For (1+x)ⁿ: coefficients are 1, n, n(n-1)/2, ... — directly from Pascal's Triangle row n
•For (1−x)ⁿ: same coefficients but alternating signs: +,−,+,−,...

CBSE marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 6-8 marks

Question type	Marks each	Typical count	What it tests
Short Answer	2	1	Expanding small powers, sum of coefficients
Long Answer	4-6	1	Finding general term, middle term, coefficient of specific power

Prep strategy

Memorise the general term formula Tᵣ₊₁ = nCᵣ·aⁿ⁻ʳ·bʳ and practise applying it with different values of a, b, and n
For 'find the term independent of x' questions: write the general term, set the power of x to zero, and solve for r
Verify middle term calculations by confirming the powers of a and b in the term add up to n

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Probability Distributions

The Binomial probability distribution P(X=r) = nCᵣ·pʳ·(1−p)ⁿ⁻ʳ directly uses the Binomial Theorem — giving the probability of r successes in n independent trials.

Approximation in Physics and Engineering

When x is very small, (1+x)ⁿ ≈ 1+nx (first two terms of the expansion) — used to approximate expressions in mechanics, optics, and error analysis.

Computer Science — Combinatorial Algorithms

The number of subsets of a set of size n is 2ⁿ (sum of all nCᵣ) — the time complexity of certain brute-force algorithms in computer science.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

For every binomial expansion question, write the general term Tᵣ₊₁ = nCᵣ·aⁿ⁻ʳ·bʳ first — it solves ALL sub-types
When a or b has a coefficient (e.g., (2x)ⁿ⁻ʳ), expand the power fully: (2x)⁴ = 16x⁴, not 2x⁴
For middle term questions: state n, determine if n is even or odd, then compute the appropriate term
Check your general term by verifying that powers of a and b sum to n

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Binomial coefficients mod prime: Lucas' theorem gives nCr mod p efficiently — important in number theory competitions
Multinomial theorem: extension to (a+b+c)ⁿ = Σ n!/(p!q!r!) · aᵖbᵍcʳ where p+q+r=n

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

CBSE Class 11 BoardHigh

JEE MainVery High

JEE AdvancedHigh

AI helpers for this chapter

Generate more practice on the fly, or have the chapter explained at a different level.

⚡

Generate 5 more questions

Fresh MCQs + short-answer items based on this chapter, plus answer keys.

🧠

Explain at a different level

Re-explain the chapter at the level you actually want.

📝 My notes

Saved on this device — sign in to sync · how this works

0/600

Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

Write the general term Tᵣ₊₁, collect the total power of x, set it equal to the required power, solve for r, then substitute back into Tᵣ₊₁.

It means the power of x in that term is 0 (x⁰=1). Set the x-exponent in the general term equal to 0, solve for r, then compute Tᵣ₊₁.

Substitute x=1 (and a=b=1) in (a+b)ⁿ = Σ nCᵣ·aⁿ⁻ʳ·bʳ: (1+1)ⁿ = Σ nCᵣ, so 2ⁿ = nC₀+nC₁+...+nCₙ.

Practice more, ask doubts, find a tutor

CBSE Class 11 mock tests

CBSE Class 11 PYQs

Ask in Q&A

Mathematics flashcards

Related chapters

Students who read Binomial Theorem usually read these next.

Sets

The language of all mathematics begins here — with SETS. This foundational chapter covers set notation, types of sets (empty, finite, infinite, equal, subsets), Venn diagrams, and set operations (union, intersection, difference, complement). CBSE Class 11 Mathematics Chapter 1.

Relations and Functions

How are things CONNECTED? Relations and functions formalise the idea of association — from the Cartesian product of sets to the special case where every input has EXACTLY ONE output. This chapter covers domain, range, and types of functions. CBSE Class 11 Mathematics Chapter 2.

Trigonometric Functions

Sine, cosine, tangent — the ratios that relate angles to sides and UNDERLIE everything from physics to engineering. This chapter covers degree and radian measure, trigonometric functions of any angle, graphs, and key identities. CBSE Class 11 Mathematics Chapter 3.

Complex Numbers and Quadratic Equations

What is √(-1)? The answer — i — opens up an ENTIRE NEW NUMBER SYSTEM. This chapter introduces complex numbers (a + ib), their algebra, the Argand plane, and how they solve every quadratic equation. CBSE Class 11 Mathematics Chapter 4.