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The term independent of x in the expansion of the expression 1 + x + 2 x 3 3 2 x 2 - 1 3 x 9 is equal to

Options

  1. A13 63
  2. B19 45
  3. C17 54
  4. D23 36

Correct answer

C. 17 54

Step-by-step solution

Coefficient of x 0 in 1 + x + 2 x 3 3 2 x 2 - 1 3 x 9 = Coefficient of x 0 in 3 2 x 2 - 1 3 x 9 + Coefficient of x - 1 in 3 2 x 2 - 1 3 x 9 + 2 × coefficient of x - 3 in 3 2 x 2 - 1 3 x 9 For 3 2 x 2 - 1 3 x 9 → T r + 1 = C r 9 3 2 x 2 9 - r - 1 3 x r = C r 9 3 2 9 - r - 1 3 r x 18 - 3 r 18 - 3 r = 0 ⇒ r = 6 , 18 - 3 r = - 1 ⇒ 3 r = 19 18 - 3 r = - 3 ⇒ 3 r = 21 ⇒ r = 7 Required coefficient = C 9 6 3 2 3 - 1 3 6 + C 9 7 3 2 2 - 1 3 7 × 2 = 3 2 2 - 1 3 6 C 6 9 3 2 + C 7 9 · 2 · - 1 3 = 17 54

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