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Schooling · Class 11 · Mathematics · Binomial Theorem · Chapter on a page

CBSE Class 11 · Mathematics · NCERT Mathematics

Binomial Theorem: chapter on a page

A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).

About the notes: Written by Shishya's AI on 27 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.

This note's current text has no applied fact-check on record.

Mind map

Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.

Mind map of Binomial Theorem: Key ideas, Formulas and facts, Watch out forKey ideasThe expansion of (a + b)ⁿ has…Each term in the expansion has the…The coefficients ⁿC₀, ⁿC₁, ⁿC₂, ……The sum of all binomial coefficients…In the expansion of (a + b)ⁿ, the (r…If n is even, there is one middle…Binomial coefficients satisfy the…Formulas and factsBinomial expansionGeneral termBinomial coefficient formulaSum of coefficientsAlternating sumMiddle term (n even)Middle terms (n odd)Watch out forForgetting that (r + 1)th term uses…Ignoring the negative sign in…Miscounting the number of terms as n…Confusing ⁿCᵣ with ⁿPᵣErrors in finding middle terms when…Binomial Theorem
The mind map as a list
  • Key ideas
    • The expansion of (a + b)ⁿ has…
    • Each term in the expansion has the…
    • The coefficients ⁿC₀, ⁿC₁, ⁿC₂, ……
    • The sum of all binomial coefficients…
    • In the expansion of (a + b)ⁿ, the (r…
    • If n is even, there is one middle…
    • Binomial coefficients satisfy the…
  • Formulas and facts
    • Binomial expansion
    • General term
    • Binomial coefficient formula
    • Sum of coefficients
    • Alternating sum
    • Middle term (n even)
    • Middle terms (n odd)
  • Watch out for
    • Forgetting that (r + 1)th term uses…
    • Ignoring the negative sign in…
    • Miscounting the number of terms as n…
    • Confusing ⁿCᵣ with ⁿPᵣ
    • Errors in finding middle terms when…

What this chapter is about

The Binomial Theorem provides a systematic method to expand expressions of the form (a + b)ⁿ, where n is a positive integer. Before this theorem, expanding (a + b)² or (a + b)³ requires repeated multiplication, which becomes impractical for higher powers. This chapter gives you a formula that directly writes out any term of such an expansion without multiplying step by step.

Key ideas

  • The expansion of (a + b)ⁿ has exactly (n + 1) terms, starting from a term with aⁿ and ending with a term containing bⁿ.
  • Each term in the expansion has the form ⁿCᵣ × aⁿ⁻ʳ × bʳ, where r takes values 0, 1, 2, …, n.
  • The coefficients ⁿC₀, ⁿC₁, ⁿC₂, …, ⁿCₙ are called binomial coefficients and appear in Pascal's Triangle, where each entry equals the sum of the two entries directly above it.
  • The sum of all binomial coefficients in (1 + x)ⁿ equals 2ⁿ, obtained by putting x = 1.
  • In the expansion of (a + b)ⁿ, the (r + 1)th term is called the general term, written as Tᵣ₊₁ = ⁿCᵣ × aⁿ⁻ʳ × bʳ.
  • If n is even, there is one middle term at position (n/2 + 1). If n is odd, there are two middle terms at positions (n + 1)/2 and (n + 3)/2.
  • Binomial coefficients satisfy the property ⁿCᵣ = ⁿCₙ₋ᵣ, which explains why coefficients are symmetric about the middle of the expansion.

Formulas and facts to remember

  • Binomial expansion: (a + b)ⁿ = ⁿC₀ aⁿ + ⁿC₁ aⁿ⁻¹ b + ⁿC₂ aⁿ⁻² b² + … + ⁿCₙ bⁿ — the complete expansion for any positive integer n.
  • General term: Tᵣ₊₁ = ⁿCᵣ × aⁿ⁻ʳ × bʳ — gives the (r + 1)th term directly without expanding everything.
  • Binomial coefficient formula: ⁿCᵣ = n! / [r! × (n − r)!] — the number of ways to choose r objects from n.
  • Sum of coefficients: ⁿC₀ + ⁿC₁ + ⁿC₂ + … + ⁿCₙ = 2ⁿ — put a = 1, b = 1 in the expansion.
  • Alternating sum: ⁿC₀ − ⁿC₁ + ⁿC₂ − ⁿC₃ + … = 0 — put a = 1, b = −1 in the expansion.
  • Middle term (n even): The middle term is T₍ₙ/₂₎₊₁.
  • Middle terms (n odd): The two middle terms are T₍ₙ₊₁₎/₂ and T₍ₙ₊₃₎/₂.

Common mistakes

  • Forgetting that (r + 1)th term uses r in the formula → always set r + 1 equal to the term number you want, then solve for r.
  • Ignoring the negative sign in expressions like (a − b)ⁿ → substitute b as (−b) and track the sign through the power.
  • Miscounting the number of terms as n instead of (n + 1) → remember that r goes from 0 to n, giving (n + 1) values.
  • Confusing ⁿCᵣ with ⁿPᵣ → binomial coefficients use combinations (order does not matter), not permutations.
  • Errors in finding middle terms when n is odd → recognise that odd n gives two middle terms, not one.

Quick revision

  • (a + b)ⁿ expands into (n + 1) terms with coefficients from Pascal's Triangle.
  • General term: Tᵣ₊₁ = ⁿCᵣ × aⁿ⁻ʳ × bʳ.
  • Sum of all binomial coefficients equals 2ⁿ.
  • ⁿCᵣ = ⁿCₙ₋ᵣ makes coefficients symmetric.
  • Even n has one middle term; odd n has two.