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JEE Main 2016

JEE Main 2016 Mathematics question on Binomial Theorem

If the number of terms in the expansion of (1−2x+4x2)n\left(1 - \dfrac{2}{x} + \dfrac{4}{x^2}\right)^n, x≠0x \neq 0, is 28, then the sum of the coefficients of all the terms in this expansion, is
  1. A.
    64
  2. B.
    2187
  3. C.
    243
  4. D.
    729
Show answer and solution

Correct option: D

Solution

The expression (1 - 2/x + 4/x²)ⁿ is a trinomial, and when expanded it has (n+1)(n+2)/2 distinct terms. Setting this equal to 28: (n+1)(n+2)/2 = 28 → (n+1)(n+2) = 56 = 7×8, so n = 6.

The sum of all coefficients is obtained by substituting x = 1: (1 - 2 + 4)⁶ = 3⁶ = 729.

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