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The coefficient of x 8 in the expansion of 1 + x 2 2 ! + x 4 4 ! + x 6 6 ! + x 8 8 ! 2 is

Options

  1. A1 315
  2. B2 315
  3. C1 105
  4. D1 210

Correct answer

A. 1 315

Step-by-step solution

Coefficient of x 8 in 1 + x 2 2 ! + x 4 4 ! + x 6 6 ! + x 8 8 ! 1 + x 2 2 ! + x 4 4 ! + x 6 6 ! + x 8 8 ! = 1 × 1 8 ! + 1 2 ! × 1 6 ! + 1 4 ! × 1 4 ! + 1 6 ! × 1 2 ! + 1 8 ! × 1 = 1 0 ! 8 ! + 1 2 ! 6 ! + 1 4 ! 4 ! + 1 6 ! 2 ! + 1 8 ! 0 ! = 1 8 ! 8 ! 0 ! 8 ! + 8 ! 2 ! 6 ! + 8 ! 4 ! 4 ! + 8 ! 6 ! 2 ! + 8 ! 8 ! 0 ! = 1 8 ! 8 C 0 + 8 C 2 + 8 C 4 + 8 C 6 + 8 C 8 = 2 8 - 1 8 ! = 2 7 8 ! = 1 315

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