India's #1 AI Tutorformula-sheet · Mathematics · Chapter 7

Class 11 Mathematics Chapter 7 Binomial Theorem — Formulas & Key Points

Q: What is the difference between the term number and r in the general term formula?

In the general term Tr+1 = nCr aⁿ⁻ʳ bʳ, the subscript r+1 indicates the term position (1st, 2nd, 3rd, etc.), while r is the value used in the binomial coefficient and powers. For the 5th term, you set r+1 = 5, giving r = 4. This is the most common mistake in CBSE Class 11 board exams — always use r = (term number - 1).

Q: How do I quickly identify the middle term when n is 11 or 12?

Check if n is even or odd. For n=12 (even), one middle term at position (12/2 + 1) = 7th term, use r=6. For n=11 (odd), two middle terms at positions (11+1)/2 = 6th and (11+3)/2 = 7th terms, use r=5 and r=6 respectively. The rule 'EVEN-ONE, ODD-TWO' helps remember this instantly during exams.

Q: Why does the alternating sum of binomial coefficients equal zero?

Set a=1 and b=-1 in the binomial theorem: (1-1)ⁿ = nC0(1)ⁿ - nC1(1)ⁿ⁻¹(1) + nC2(1)ⁿ⁻²(1)² -... This simplifies to 0 = nC0 - nC1 + nC2 - nC3 +..., proving the alternating sum is zero. This property appears in coefficient sum problems worth 2-3 marks in CBSE Class 11 Mathematics papers.

Q: How can I find a specific coefficient without expanding the entire binomial?

Use the general term formula Tr+1 = nCr aⁿ⁻ʳ bʳ. Identify which term contains your target variable power by setting the power expression equal to your requirement and solving for r. Then calculate that single term's coefficient using nCr and the numerical factors from a and b. This saves time compared to full expansion in board exams.

Q: What is Pascal's triangle and how does it help with binomial expansion?

Pascal's triangle arranges binomial coefficients in rows where row n contains the coefficients for (a+b)ⁿ. Each number equals the sum of the two numbers above it. Row 4 is [1, 4, 6, 4, 1], giving coefficients for (a+b)⁴ = 1a⁴ + 4a³b + 6a²b² + 4ab³ + 1b⁴. For small n (up to 6-7), Pascal's triangle provides coefficients faster than calculating nCr individually.

Q: How do I handle negative terms in (a - b)ⁿ expansion correctly?

Write the general term as Tr+1 = nCr aⁿ⁻ʳ (-b)ʳ = (-1)ʳ nCr aⁿ⁻ʳ bʳ. The factor (-1)ʳ ensures correct signs: positive when r is even, negative when r is odd. Many Class 11 students forget this factor and lose 1-2 marks. Always include (-1)ʳ explicitly in your working when dealing with (a - b)ⁿ or similar expressions with subtraction.

Q: What marks weightage does Binomial Theorem carry in CBSE Class 11 final exam?

CBSE typically allocates 6-8 marks to Binomial Theorem in the Class 11 Mathematics final exam, usually as 2-3 questions: one short answer (2-3 marks) finding a specific term or coefficient, one long answer (4-5 marks) involving middle term or coefficient properties, and occasionally a 1-mark multiple-choice question. The chapter appears in both Section B and Section C of the paper.

Q: Can the binomial theorem be applied when n is negative or a fraction?

The NCERT Class 11 Mathematics Chapter 7 binomial theorem requires n to be a non-negative integer (0, 1, 2, 3,...). For negative or fractional n, a generalized binomial theorem exists using infinite series, but this is covered in Class 12 or higher mathematics courses. In your Class 11 syllabus, always assume n is a positive integer unless explicitly stated otherwise.

Q: How does CBSETUTOR.ai help with binomial theorem problem-solving?

CBSETUTOR.ai offers 24×7 AI tutoring where Class 11 students can upload photos of any binomial theorem problem — whether finding general terms, middle terms, or coefficient sums — and receive step-by-step solutions instantly. The platform covers all NCERT Class 11 Mathematics chapters at ₹999/month for any class (6-12) with a 3-day free trial, eliminating the need for expensive city coaching for binomial expansion practice.

Q: What is the fastest way to calculate nCr during an exam without a calculator?

Use the formula nCr = n!/(r!(n-r)!) but simplify before calculating. Choose the smaller of r or (n-r) due to symmetry: 10C7 = 10C3 is easier. Write out partial factorials: 10C3 = (10×9×8)/(3×2×1) = 720/6 = 120. Cancel common factors before multiplying to avoid large numbers. Memorize common values like 5C2=10, 6C3=20, 7C3=35, 10C5=252 for quick recall during CBSE exams.

CBSE Class 11 Mathematics Chapter 7 Binomial Theorem is central to algebra and combinatorics, appearing in 6-8 marks worth of questions in board exams. This formula sheet consolidates every expansion identity, the general term formula Tr+1, middle term rules, binomial coefficient properties, and Pascal's triangle patterns from your NCERT Class 11 Mathematics textbook into one revision resource with memory aids and worked examples.

Your child's private AI tutor — trained on NCERT.

3-day free trial · ₹1 to start · Cancel anytime.

Start 3-day free trial →

Key takeaways

✓The binomial theorem states (a + b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ for r from 0 to n, valid for positive integer n.
✓General term formula is Tr+1 = nCr an-r br where r ranges from 0 to n; the (r+1)th term has r in the combination.
✓Middle term depends on n: if n is even, one middle term at position (n/2 + 1); if n is odd, two middle terms at positions (n+1)/2 and (n+3)/2.
✓Binomial coefficients nC0, nC1, nC2,... nCn form row n of Pascal's triangle; each coefficient equals nCr = n! / (r!(n-r)!).
✓For (a - b)ⁿ expansion, alternate signs appear: positive terms when r is even, negative when r is odd.
✓Greatest binomial coefficient in (a + b)ⁿ is nCn/2 when n is even, or nC(n-1)/2 = nC(n+1)/2 when n is odd.
✓Applications include finding specific terms without full expansion, determining coefficients of particular variables, and approximation calculations.

Core Binomial Theorem Formulas

The binomial theorem provides the algebraic expansion of (a + b) raised to any positive integer power n. These formulas form the foundation of the chapter and appear in nearly every Class 11 Mathematics board question on this topic. Understanding when each form applies — whether positive binomial (a + b)ⁿ, negative binomial (a - b)ⁿ, or special cases with coefficients — is crucial for solving problems efficiently. NCERT Class 11 Mathematics presents these in progressive complexity starting from simple squares and cubes.

General Term Formula and Position Rules

The general term formula is the most frequently tested concept in CBSE Class 11 Mathematics Chapter 7. The (r+1)th term in the expansion of (a + b)ⁿ is given by Tr+1 = nCr aⁿ⁻ʳ bʳ. Notice carefully: the term number is r+1 but the combination uses r. This formula allows you to find any specific term without expanding the entire binomial. Students often confuse the subscript notation — Tr+1 means the term at position r+1, where r starts from 0. For finding the 5th term, set r+1 = 5, so r = 4. The powers of a and b always sum to n in any term.

General term: Tr+1 = nCr aⁿ⁻ʳ bʳ where r = 0, 1, 2,..., n
First term (r=0): T1 = nC0 aⁿ = aⁿ
Second term (r=1): T2 = nC1 aⁿ⁻¹ b = n·aⁿ⁻¹·b
Last term (r=n): Tn+1 = nCn bⁿ = bⁿ
For (a - b)ⁿ: Tr+1 = (-1)ʳ nCr aⁿ⁻ʳ bʳ
Total number of terms in expansion = n + 1

Middle Term Formulas and Identification

Middle term questions carry 3-4 marks in CBSE board exams and require careful identification based on whether n is even or odd. When n is even, there is exactly one middle term at position (n/2 + 1). When n is odd, there are two middle terms at positions (n+1)/2 and (n+3)/2. This distinction is critical — many Class 11 Mathematics students lose marks by applying the wrong formula. The middle term often has the greatest binomial coefficient and appears in optimization problems. Remember that position refers to term number, not the value of r in the general term formula.

Binomial Coefficient Properties and Pascal's Triangle

Binomial coefficients nC0, nC1, nC2,... nCn have elegant symmetry and summation properties that simplify calculations in CBSE Class 11 Mathematics problems. These coefficients form Pascal's triangle where each number is the sum of the two numbers directly above it. The property nCr = nCn-r reflects the symmetry of the triangle. The sum of all binomial coefficients in (a + b)ⁿ equals 2ⁿ, found by setting a = b = 1. For alternating sums, setting a = 1 and b = -1 gives zero. Understanding these properties helps solve coefficient-sum problems without tedious calculation of individual terms.

Symmetry property: nCr = nCn-r for all r from 0 to n
Sum of coefficients: nC0 + nC1 + nC2 +... + nCn = 2ⁿ
Alternating sum: nC0 - nC1 + nC2 - nC3 +... + (-1)ⁿ nCn = 0
Pascal's identity: nCr + nCr-1 = n+1Cr
Greatest coefficient when n even: nCn/2
Greatest coefficient when n odd: nC(n-1)/2 = nC(n+1)/2

Key Terms and Definitions

Mastering the precise terminology from NCERT Class 11 Mathematics Chapter 7 ensures you write accurate solutions in board exams. Binomial means 'two terms' — expressions like (a + b), (x - y), or (2p + 3q). The expansion refers to writing the power as a sum of terms. Index or exponent is the power n to which the binomial is raised, always a non-negative integer in this chapter. Binomial coefficients are the numerical factors nCr in each term. General term provides a formula for any term without full expansion. Middle term is the term(s) equidistant from both ends of the expansion. These definitions appear verbatim in CBSE marking schemes.

Important Values and Standard Results

Certain binomial expansions and coefficient values appear so frequently in Class 11 Mathematics that memorizing them saves time in exams. The expansions of (a ± b)² and (a ± b)³ are algebraic identities you have known since Class 9, but recognizing them as binomial theorem applications with n=2 and n=3 deepens understanding. The sum 2ⁿ for total coefficients, 0 for alternating sum, and specific values like 10C5 = 252 are worth remembering. When calculating approximations, (1 + x)ⁿ ≈ 1 + nx for small |x| gives quick estimates. CBSETUTOR.ai provides 24×7 AI tutoring where students can photograph any binomial theorem problem and receive step-by-step solutions with these standard results applied contextually, all for ₹999/month across Classes 6-12 with a 3-day free trial.

(a + b)² = a² + 2ab + b² [coefficients: 1, 2, 1]
(a - b)² = a² - 2ab + b² [coefficients: 1, -2, 1]
(a + b)³ = a³ + 3a²b + 3ab² + b³ [coefficients: 1, 3, 3, 1]
(1 + x)ⁿ ≈ 1 + nx when |x| << 1 (approximation formula)
Sum of binomial coefficients: 2ⁿ for (a+b)ⁿ
Some useful values: 5C2 = 10, 6C3 = 20, 7C3 = 35, 10C5 = 252

Memory Tricks and Mnemonics

Class 11 Mathematics students preparing for CBSE boards find these memory devices helpful for quick recall during exams. For the general term formula Tr+1 = nCr aⁿ⁻ʳ bʳ, remember 'GR-RAIS': General term has R in Combination, R in b's power, And Index minus r in a's power, Subscript r+1. For middle term identification, the mnemonic 'EVEN-ONE, ODD-TWO' reminds you that even n gives one middle term, odd n gives two. The symmetry nCr = nCn-r can be remembered as 'mirror coefficients' — Pascal's triangle is perfectly symmetrical. When signs alternate in (a - b)ⁿ, think 'Even r Positive, Odd r Negative' for the term (-1)ʳ.

GR-RAIS for general term: General R in Combination, R in b's power, And Index-R in a's power, Subscript r+1
EVEN-ONE, ODD-TWO: Even n has one middle term, odd n has two middle terms
POPS for power check: Powers Of a Plus powers of b = n in every term
Mirror Coefficients: nCr = nCn-r, Pascal's triangle is symmetrical
EPON for signs in (a-b)ⁿ: Even r Positive, Odd r Negative for (-1)ʳ
Total terms memory: 'n plus one' — expansion of (a+b)ⁿ always has n+1 terms

Common Mistakes in Notation and Signs

CBSE Class 11 Mathematics examiners consistently report these errors in binomial theorem answers. The most frequent mistake is confusing term position with r value — for the 5th term, students write T5 but then incorrectly use r=5 instead of r=4 in the general term formula. Sign errors occur when expanding (a - b)ⁿ because students forget to include (-1)ʳ or miscalculate when r is odd versus even. Writing nCr as ⁿCr or nCʳ (with superscripts in wrong places) loses marks for notation. Another common error is writing n!/(r!(n-r)!) incorrectly as n!/(r!n-r!) without parentheses. In exam pressure, students sometimes add powers instead of ensuring they sum to n, writing terms like a⁵b⁶ in a (a+b)⁹ expansion.

Term position error: For Tk, use r = k-1 in formula, not r = k
Sign mistake: In (a-b)ⁿ, include (-1)ʳ factor; negative when r is odd
Notation: Write nCr clearly, not ⁿCr, Cⁿr, or nCʳ
Factorial: n!/(r!(n-r)!) needs parentheses around (n-r), not n!/(r!n-r!)
Power sum: In every term, power of a plus power of b must equal n
Coefficient confusion: nCr ≠ n×r; it is n!/(r!(n-r)!), not simple multiplication

Three Solved Mini-Examples

These worked problems demonstrate typical CBSE Class 11 Mathematics board exam patterns for binomial theorem questions. Example 1 tests finding a specific term using general term formula. Example 2 involves identifying and calculating the middle term when n is even. Example 3 applies binomial coefficient properties to find sum of coefficients. Each solution follows the step-by-step method expected in CBSE marking schemes, showing formula identification, substitution, calculation, and final answer clearly. Practice these patterns to build confidence for your Class 11 Mathematics solutions.

One-Glance Last-Minute Revision Box

This ultra-compact summary is designed for final revision 30 minutes before your CBSE Class 11 Mathematics exam. Cover the core formulas, middle term rules, coefficient properties, and common pitfalls in one visual sweep. Print this section or photograph it on your phone for quick reference during study sessions. Every formula here is board-exam tested and appears in NCERT Class 11 Mathematics Chapter 7 exercises. Pair this revision box with solving previous year CBSE papers for maximum retention and exam confidence.

Frequently asked questions

What is the difference between the term number and r in the general term formula?+

In the general term Tr+1 = nCr aⁿ⁻ʳ bʳ, the subscript r+1 indicates the term position (1st, 2nd, 3rd, etc.), while r is the value used in the binomial coefficient and powers. For the 5th term, you set r+1 = 5, giving r = 4. This is the most common mistake in CBSE Class 11 board exams — always use r = (term number - 1).

How do I quickly identify the middle term when n is 11 or 12?+

Check if n is even or odd. For n=12 (even), one middle term at position (12/2 + 1) = 7th term, use r=6. For n=11 (odd), two middle terms at positions (11+1)/2 = 6th and (11+3)/2 = 7th terms, use r=5 and r=6 respectively. The rule 'EVEN-ONE, ODD-TWO' helps remember this instantly during exams.

Why does the alternating sum of binomial coefficients equal zero?+

Set a=1 and b=-1 in the binomial theorem: (1-1)ⁿ = nC0(1)ⁿ - nC1(1)ⁿ⁻¹(1) + nC2(1)ⁿ⁻²(1)² -... This simplifies to 0 = nC0 - nC1 + nC2 - nC3 +..., proving the alternating sum is zero. This property appears in coefficient sum problems worth 2-3 marks in CBSE Class 11 Mathematics papers.

How can I find a specific coefficient without expanding the entire binomial?+

Use the general term formula Tr+1 = nCr aⁿ⁻ʳ bʳ. Identify which term contains your target variable power by setting the power expression equal to your requirement and solving for r. Then calculate that single term's coefficient using nCr and the numerical factors from a and b. This saves time compared to full expansion in board exams.

What is Pascal's triangle and how does it help with binomial expansion?+

Pascal's triangle arranges binomial coefficients in rows where row n contains the coefficients for (a+b)ⁿ. Each number equals the sum of the two numbers above it. Row 4 is [1, 4, 6, 4, 1], giving coefficients for (a+b)⁴ = 1a⁴ + 4a³b + 6a²b² + 4ab³ + 1b⁴. For small n (up to 6-7), Pascal's triangle provides coefficients faster than calculating nCr individually.

How do I handle negative terms in (a - b)ⁿ expansion correctly?+

Write the general term as Tr+1 = nCr aⁿ⁻ʳ (-b)ʳ = (-1)ʳ nCr aⁿ⁻ʳ bʳ. The factor (-1)ʳ ensures correct signs: positive when r is even, negative when r is odd. Many Class 11 students forget this factor and lose 1-2 marks. Always include (-1)ʳ explicitly in your working when dealing with (a - b)ⁿ or similar expressions with subtraction.

What marks weightage does Binomial Theorem carry in CBSE Class 11 final exam?+

CBSE typically allocates 6-8 marks to Binomial Theorem in the Class 11 Mathematics final exam, usually as 2-3 questions: one short answer (2-3 marks) finding a specific term or coefficient, one long answer (4-5 marks) involving middle term or coefficient properties, and occasionally a 1-mark multiple-choice question. The chapter appears in both Section B and Section C of the paper.

Can the binomial theorem be applied when n is negative or a fraction?+

The NCERT Class 11 Mathematics Chapter 7 binomial theorem requires n to be a non-negative integer (0, 1, 2, 3,...). For negative or fractional n, a generalized binomial theorem exists using infinite series, but this is covered in Class 12 or higher mathematics courses. In your Class 11 syllabus, always assume n is a positive integer unless explicitly stated otherwise.

How does CBSETUTOR.ai help with binomial theorem problem-solving?+

CBSETUTOR.ai offers 24×7 AI tutoring where Class 11 students can upload photos of any binomial theorem problem — whether finding general terms, middle terms, or coefficient sums — and receive step-by-step solutions instantly. The platform covers all NCERT Class 11 Mathematics chapters at ₹999/month for any class (6-12) with a 3-day free trial, eliminating the need for expensive city coaching for binomial expansion practice.

What is the fastest way to calculate nCr during an exam without a calculator?+

Use the formula nCr = n!/(r!(n-r)!) but simplify before calculating. Choose the smaller of r or (n-r) due to symmetry: 10C7 = 10C3 is easier. Write out partial factorials: 10C3 = (10×9×8)/(3×2×1) = 720/6 = 120. Cancel common factors before multiplying to avoid large numbers. Memorize common values like 5C2=10, 6C3=20, 7C3=35, 10C5=252 for quick recall during CBSE exams.

Related resources

CBSE Class 11 Mathematics Chapter 7 Binomial Theorem Worksheet with Answers Important Questions: CBSE Class 11 Mathematics Chapter 7 Binomial Theorem Important Questions: CBSE Class 11 Mathematics Chapter 6 Permutations and Combinations Class 11 Mathematics Chapter 6 Permutations and Combinations — Formulas & Key Points AI Tutor for Class 11: The Smart Alternative to Tuition AI Tutor for Class 11 Accountancy: Learn Faster with Instant Help Class 9 Coordinate Geometry Chapter 3: Cartesian System & Plotting Points Class 9 Mathematics Chapter 2 Polynomials — Formulas & Key Points

Keep learning — related guides

Class 11Mathematics

Class 11 Mathematics Chapter 8 Sequences and Series — Formulas & Key Points

Class 11Mathematics

Important Questions: CBSE Class 11 Mathematics Chapter 14 Probability

Class 11Mathematics

CBSE Class 11 Mathematics Chapter 14 Probability Worksheet with Answers

Class 11Mathematics

Class 11 Mathematics Chapter 14 Probability — Formulas & Key Points

Class 11Mathematics

Important Questions: CBSE Class 11 Mathematics Chapter 13 Statistics

Class 11Mathematics

CBSE Class 11 Mathematics Chapter 13 Statistics Worksheet with Answers

Ready to give your Class 11 child the tutor that never sleeps?

CBSETUTOR.ai covers every chapter in the Class 11 NCERT syllabus — Maths, Science, Social Science, English, Hindi and more. 24×7. Patient. Unlimited. 3-day free trial.

Start your child's 3-day free trial →