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Class 11 · Chapter 7 · Algebra unit (25 of 80 marks)

Binomial Theorem Class 11: notes and important questions

Revision notes, 17 board-style questions with the step marks shown, and a four-step route from the basics to 95+, each step ending in a short checkpoint.

  • 17 questions
  • 5 multiple choice, 1 assertion–reason, 3 very short answer, 4 short answer, 2 long answer, 2 case study
  • About 7 hours to master

Algebra unit: 25 of 80 theory marks (Complex Numbers and Quadratic Equations, Linear Inequalities, Permutations and Combinations, Binomial Theorem, Sequences and Series).

Revision notes

Binomial Theorem: revision notes

1. Pascal's triangle

The coefficients of \((a + b)^n\) for \(n = 0, 1, 2, \ldots\) form Pascal's triangle: each row starts and ends with \(1\), and every other entry is the sum of the two above it.

\(1\); \(1\ 1\); \(1\ 2\ 1\); \(1\ 3\ 3\ 1\); \(1\ 4\ 6\ 4\ 1\); \(1\ 5\ 10\ 10\ 5\ 1\); \(1\ 6\ 15\ 20\ 15\ 6\ 1\). Entry \(r\) of row \(n\) is \({}^nC_r\) (because \({}^nC_r + {}^nC_{r-1} = {}^{n+1}C_r\)).

2. The binomial theorem

For a positive integer \(n\):

\[(a + b)^n = {}^nC_0a^n + {}^nC_1a^{n-1}b + {}^nC_2a^{n-2}b^2 + \cdots + {}^nC_nb^n\]

  • There are \(n + 1\) terms; powers of \(a\) fall from \(n\) to \(0\) while powers of \(b\) rise from \(0\) to \(n\); in each term the powers add up to \(n\).
  • For \((a - b)^n\) the signs alternate: \(+, -, +, \ldots\)
  • \(a = b = 1\): \({}^nC_0 + {}^nC_1 + \cdots + {}^nC_n = 2^n\). Putting \(x = 1\) in any expansion gives the sum of its coefficients.

3. Special cases

\((1 + x)^n = 1 + nx + \dfrac{n(n - 1)}{2}x^2 + \cdots + x^n\). \((a + b)^n + (a - b)^n\) keeps only the terms with even powers of \(b\) (doubled); \((a + b)^n - (a - b)^n\) only the odd ones.

4. Applications

  • Powers of numbers near a round number: \(101^3 = (100 + 1)^3\), \((0.99)^4 = (1 - 0.01)^4\).
  • Divisibility: \((1 + k)^n = 1 + nk + k^2(\ldots)\), so \((1 + k)^n - 1 - nk\) is divisible by \(k^2\).
  • Comparing sizes: for \(x \gt 0\), \((1 + x)^n \gt 1 + nx\) (\(n \ge 2\)).

Worked example 1

Expand \((1 + 2x)^3\).

\(1 + 3(2x) + 3(2x)^2 + (2x)^3 = 1 + 6x + 12x^2 + 8x^3\).

Worked example 2

Evaluate \(99^3\) using the binomial theorem.

\((100 - 1)^3 = 1000000 - 30000 + 300 - 1 = 970299\).

Worked example 3

Find \((\sqrt2 + 1)^4 + (\sqrt2 - 1)^4\).

\(2[(\sqrt2)^4 + 6(\sqrt2)^2 + 1] = 2(4 + 12 + 1) = 34\).

Common errors

  • Forgetting to raise the coefficient of \(x\) to the power: \((2x)^3 = 8x^3\), not \(2x^3\).
  • Signs in \((a - b)^n\): every odd power of \(b\) carries a minus sign.
  • Using row \(n - 1\) of Pascal's triangle for \((a + b)^n\) (the row starting \(1, n\) is the right one).
  • Writing \(n\) terms instead of \(n + 1\).

Exam tips

  • Write the coefficients \(1, n, {}^nC_2, \ldots\) first, then the powers of each part in brackets.
  • Check an expansion by putting \(x = 1\): both sides must agree.

Topics in this chapter: The binomial theorem · Pascal's triangle · Special cases · Applications.

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

You can use Pascal's triangle, count terms and find coefficients in small expansions.

Read first: 1. Pascal's triangle; 2. The binomial theorem 5 practice questions · checkpoint: 3 questions, 3 marks, pass 80%
Practise step 1
Step 2

Exam standard

You can expand binomials with coefficients and fractions and evaluate powers of numbers near round numbers.

Read first: 2. The binomial theorem; 4. Applications; Worked examples 1-3 6 practice questions · checkpoint: 3 questions, 8 marks, pass 80%
Practise step 2
Step 3

Full marks on long answers

You can prove divisibility results and combine (a + b)ⁿ ± (a − b)ⁿ in multi-part questions.

Read first: 3. Special cases; 4. Divisibility 3 practice questions · checkpoint: 2 questions, 9 marks, pass 80%
Practise step 3
Step 4

95+ stretch (HOTS)

You can compare sizes of powers and reason about sums of coefficients.

Read first: 4. Comparing sizes; 2. Sum of coefficients 3 practice questions · checkpoint: 3 questions, 9 marks, pass 80%
Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 2 of the 17 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceThe binomial theorem

The number of terms in the expansion of \((a + b)^7\) is

  1. (a)\(7\)
  2. (b)\(8\)
  3. (c)\(14\)
  4. (d)\(6\)
Q2·1 mark·Multiple choicePascal's triangle

The coefficient of \(a^2b^3\) in \((a + b)^5\) is

  1. (a)\(5\)
  2. (b)\(20\)
  3. (c)\(15\)
  4. (d)\(10\)
Q3·1 mark·Multiple choiceThe binomial theorem

The sum of the coefficients in the expansion of \((1 + x)^6\) is

  1. (a)\(6\)
  2. (b)\(36\)
  3. (c)\(64\)
  4. (d)\(32\)

Stretch yourself: 7 competition-style combinatorics problems (Senior, ages 16 to 18), original, with hints and full solutions. More extension & competition maths →

Where marks are lost in Binomial Theorem

  • Coefficient inside the bracket not raised to the power: (3x)² written 3x². Fix: bracket every term before raising it.
  • Alternating signs missed in (a − b)ⁿ. Fix: the sign is (−1)^(power of b).
  • Wrong row of Pascal's triangle. Fix: (a + b)ⁿ uses the row 1, n, …, n, 1 with n + 1 entries.
  • Divisibility proofs that stop after expanding. Fix: take the common factor out and say what is left is an integer.
  • Size comparisons claimed without an inequality. Fix: keep the first three terms and show they already exceed the other number.

Test Binomial Theorem against the clock

Want it against the clock? Take a timed 30-mark chapter test on Binomial Theorem, new questions each time (CBSE Essentials or the free trial).