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Manipal MET2024MathematicsBinomial Theorem

If the coefficient of x^m, m N in the expansion of ( 2 x + [3] 3 x^2 )^9 is equal to k(k 0) , the k is equal to

Options

  1. A1008
  2. B2016
  3. C3024
  4. D1016

Correct answer

B. 2016

Step-by-step solution

aligned T_ r+1 & = ^9 C_r(2 x)^ 9-r 2 ( 3 x^2 )^ r 3 & = ^9 C_r 2^ 9-r 2 3^ r 3 x^ ( 9-r 2 - 2 3 ) & = ^9 C_r 2^ 9-r 2 3^ r 3 x^ 27-7 r 6 aligned So, r=3 aligned & Then, coefficient = ^9 C₃ 2^ 9-3 2 3^1 & aligned & = 9 8 7 3 2 2^3 3 & =2016 aligned aligned

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