JEE JEE Main and JEE Advanced · Mathematics

Binomial Theorem

Expand a binomial with a positive integral index, identify a general or middle term, and use binomial-coefficient properties without drifting into generalized binomial-series claims.

Subject
Mathematics
Syllabus unit
Binomial Theorem
  • Positive integral index only: no generalized binomial series
  • General term and middle-term position depend on parity of n
  • No invented weightage, question counts or trend percentages

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In short

For a positive integer n, (a+b)^n has n+1 terms and its (r+1)th term is C(n,r)a^(n-r)b^r, where 0≤r≤n.

The number and location of middle terms depend on whether n is even or odd.

Syllabus mapping

  • Unit
    Binomial Theorem
    Topics
    Binomial theorem for a positive integral index, General term, Middle term, Simple applications, Properties of binomial coefficients

What this chapter contains and why it matters

  • Question
    What is the chapter about?
    Direct answer
    Expanding a binomial with a positive integral index, finding general and middle terms, and using coefficient properties.
  • Question
    What is the central method choice?
    Direct answer
    Confirm the index is a positive integer, convert the requested term position to r, and check parity of n for middle terms.
  • Question
    Where do most mistakes begin?
    Direct answer
    Using the theorem for a non-integral or negative index, confusing r with the term number r+1, and choosing one middle term for odd n.
  • Question
    What should come before Binomial Theorem?
    Direct answer
    Permutations and Combinations, for the combinations notation used in coefficients.
  • Question
    What comes after it?
    Direct answer
    Probability builds on counting and coefficient reasoning in later problems.

The official JEE documents define content scope. They do not publish chapter weightage, so none is asserted here.

Official JEE syllabus mapping for Binomial Theorem

Verified against the current JEE Main 2026 syllabus and JEE Advanced 2026 syllabus.

  • Concept group
    Positive integral index expansion
    JEE Main 2026
    Explicitly limits the unit wording to Binomial Theorem for a positive integral index, the general term, middle term and simple applications.
    JEE Advanced 2026
    Explicitly includes Binomial Theorem for a positive integral index and properties of binomial coefficients.

Sources: JEE Main 2026 syllabus and JEE Advanced 2026 syllabus, both linked in the sources section below.

Before this chapter

Concepts in this chapter

1. Confirm the index is a positive integer

The theorem used on this page requires n to be a positive integer.

Confirm n is a positive integer for the official theorem used here. A non-integral or negative index is not part of this route's scope.

2. Convert term position to r

T_(r+1) uses coefficient index r, so the requested term number is one more than r.

Remember that T_(r+1) uses coefficient index r. Count total terms as n+1, and check that r lies between 0 and n before simplifying.

3. Determine parity of n before choosing the middle term

n even gives one middle term; n odd gives two middle terms.

Determine parity of n before selecting middle term positions. For n even the middle term is T_(n/2+1). For n odd the middle terms are T_((n+1)/2) and T_((n+3)/2).

4. Use coefficient properties for identities

Symmetry, Pascal's identity, and substitution into (1+x)^n give coefficient results without full expansion.

Coefficient symmetry, Pascal's identity, and substitution into a general expansion allow coefficient sums and comparisons to be found without expanding the whole binomial.

Method selector

  • Question signal
    Full small expansion
    First model
    Apply the theorem directly.
  • Question signal
    One requested term
    First model
    Use the general term without expanding everything.
  • Question signal
    Term independent of a variable
    First model
    Set that variable's resulting exponent to zero and solve for integer r in range.
  • Question signal
    Middle term
    First model
    Check parity of n first.
  • Question signal
    Coefficient identity
    First model
    Use symmetry, Pascal identity or substitution into (1+x)^n as appropriate.

Formula sheet

  • Sum from r equals zero to n of C n r times a to the power n minus r times b to the power r.

    Expansion of a binomial raised to a positive integer power.

    n
    positive integer index
    r
    coefficient index
    C(n,r)
    binomial coefficient, n!/[r!(n-r)!]

    Use when — Full expansion is required.

    Common trap — Importing generalized-binomial conditions for non-integer n.

  • T sub r plus one equals C n r times a to the power n minus r times b to the power r.

    The (r+1)th term of the expansion.

    r
    coefficient index, integer with 0≤r≤n

    Use when — A specific term is requested without expanding the whole binomial.

    Common trap — Calling this term T_r instead of T_(r+1).

  • C n r equals C n, n minus r.

    Binomial coefficients equidistant from the ends are equal.

    r
    integer with 0≤r≤n

    Use when — Coefficient comparison across paired positions.

    Common trap — Using an out-of-range r.

  • C n r plus C n, r plus one equals C n plus one, r plus one.

    Adjacent coefficients of row n combine into a coefficient of row n+1.

    r
    integer with valid coefficient indices

    Use when — Coefficient transformations across rows.

    Common trap — Index shift error.

  • Sum of C n r over all r equals two to the power n.

    Sum of all binomial coefficients in row n.

    n
    positive integer

    Use when — Coefficient sum questions.

    Common trap — Forgetting all terms from r=0 to n.

  • Sum of minus one to the power r times C n r equals zero.

    Alternating sum of binomial coefficients in row n.

    n
    positive integer

    Use when — Alternating coefficient sum questions.

    Common trap — Using n=0 outside the positive-index page scope.

  • T sub n over two plus one.

    The single middle term when n is even.

    n
    even positive integer

    Use when — Finding the central term for even n.

    Common trap — Using the odd-n formula.

  • T sub n plus one over two, and T sub n plus three over two.

    The two middle terms when n is odd.

    n
    odd positive integer

    Use when — Finding the central pair for odd n.

    Common trap — Assuming a single middle term.

Worked examples

Find the middle term of (x+2)^6.

Answer: 160x^3

n=6, so total terms = 7. n is even, so use T_(n/2+1) = T_4, giving r+1=4 so r=3.

T_4 = C(6,3) x^(6-3) 2^3 = 20 x^3 × 8 = 160x^3.

Validity check: the seven term positions are 1 through 7, so T_4 is the unique central position.

Common mistakes and what they actually indicate

  • Using the theorem here for a non-integral or negative index.

    Knowledge gap

    Why it happens

    The theorem on this page is restricted to a positive integral index in the current JEE scope.

    How it is corrected

    Confirm n is a positive integer before applying any formula on this page.

  • Confusing r with the term number r+1.

    Execution error

    Why it happens

    The general term T_(r+1) is indexed by r, one less than the term number.

    How it is corrected

    Convert the requested term number to r before substituting into the formula.

  • Saying there are n terms rather than n+1.

    Recall gap

    Why it happens

    The expansion runs from r=0 to r=n, giving n+1 terms.

    How it is corrected

    Count term positions from 1 to n+1 explicitly.

  • Choosing one middle term for odd n.

    Decision / selection error

    Why it happens

    Odd n gives an even number of terms (n+1), so there is a central pair, not a single middle term.

    How it is corrected

    Check parity of n before selecting the middle-term formula.

  • Forgetting coefficient-index restrictions.

    Needs review

    Why it happens

    C(n,r) is only defined for integers with 0≤r≤n.

    How it is corrected

    Check the range of r before using any coefficient identity.

  • Adding generalized binomial-series material not supported by the current route scope.

    Needs review

    Why it happens

    The current JEE syllabus for this page is limited to a positive integral index.

    How it is corrected

    Keep this page's scope to positive integral index expansions only.

Sources and provenance

Official syllabus scope verified against NTA JEE Main 2026 and JEE Advanced 2026 documents. No weightage, frequency or PYQ-count claim is made.

Contributor requirements for this page

  • Reviewer specialisation: combinatorics and algebra.
  • Minimum: postgraduate Mathematics, Applied Mathematics or equivalent.
  • Review scope: positive-index boundary, coefficient-index conditions, middle-term formulas, coefficient identities and scope control against generalized expansion.