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JEE Main20231 Feb 2023Morning ShiftMathematicsBinomial TheoremActual

The value of 1 1 ! 50 ! + 1 3 ! 48 ! + 1 5 ! 46 ! + … . + 1 49 ! 2 ! + 1 51 ! 1 ! is

Options

  1. A2 50 50 !
  2. B2 50 51 !
  3. C2 51 51 !
  4. D2 51 50 !

Correct answer

B. 2 50 51 !

Step-by-step solution

Let S = 1 1 ! · 50 ! + 1 3 ! · 48 ! + 1 5 ! · 46 ! + . . . . + 1 51 ! · 1 ! ⇒ S = 1 51 ! 51 ! 1 ! · 50 ! + 51 ! 3 ! · 48 ! + 51 ! 5 ! · 46 ! + . . . . + 51 ! 51 ! · 0 ! ⇒ S = 1 51 ! C 1 51 + C 3 51 + C 5 51 + . . . . + C 51 51 ⇒ S = 2 51 - 1 51 ! ⇒ S = 2 50 51 !

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