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The coefficient of x 5 in the expansion of 1 + x 1 ! + x 2 2 ! + x 3 3 ! + x 4 4 ! + x 5 5 ! 2 is

Options

  1. A2 15
  2. B4 15
  3. C1 30
  4. D2 45

Correct answer

B. 4 15

Step-by-step solution

Coefficient of x 5 in 1 + x 1 ! + x 2 2 ! + x 3 3 ! + x 4 4 ! + x 5 5 ! 1 + x 1 ! + x 2 2 ! + x 3 3 ! + x 4 4 ! + x 5 5 ! = 1 × 1 5 ! + 1 1 ! × 1 4 ! + 1 2 ! × 1 3 ! + 1 3 ! × 1 3 ! × 1 2 ! + 1 4 ! × 1 1 ! + 1 5 ! × 1 = 1 5 ! 5 ! 1 ! 5 ! + 5 ! 1 ! 4 ! + 5 ! 2 ! 3 ! + 5 ! 3 ! 2 ! + 5 ! 4 ! 1 ! + 5 ! 5 ! 0 ! = 1 5 ! 5 C 0 + 5 C 1 + 5 C 2 + 5 C 3 + 5 C 4 + 5 C 5 = 2 5 5 ! = 32 120 = 4 15 .

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