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JEE Main2012MathematicsBinomial TheoremActual

The middle term in the expansion of (1- 1 x )^n (1-x^n ) in powers of x is

Options

  1. A- ^ 2 n C _ n-1
  2. B- ^ 2 n C _n
  3. C^ 2 n C _ n-1
  4. D^ 2 n C _n

Correct answer

D. ^ 2 n C _n

Step-by-step solution

Given expansion can be re-written as ( x-1 x )^n (1-x)^n=(-1)^n x^ -n (1-x)^ 2 n Total number of terms will be 2 n+1 which is odd ( 2 n is always even) Middle term = 2 n+1+1 2 =(n+1) th Now, T_ r+1 = ^n C_r(1)^r x^ n-r So, ^ 2 n C_n x^ 2 n-n x^n (-1)^n = ^ 2 n C_n (-1)^n Middle term is an odd term. So, n+1 will be odd. So, n will be even. Required answer is ^ 2 n C _n .

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