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JEE Main201910 Jan 2019Evening ShiftMathematicsBinomial TheoremActual

If ∑ r = 0 25 C r 50 C 25 - r 50 - r = K C 25 50 , then K is equal to

Options

  1. A2 25
  2. B2 25 - 1
  3. C2 24
  4. D25 2

Correct answer

A. 2 25

Step-by-step solution

Given: ∑ r = 0 25 C r   50 C 25 - r   50 - r = K C 25   50 LHS = ∑ r = 0 25   50 C r   50 - r C 25 - r = ∑ r = 0 25 50 ! r ! 50 - r ! 50 - r ! 25 - r ! 25 ! = ∑ r = 0 25   50 !    25 ! r ! 25 - r ! 25 ! 25 ! =   50 C 25   ∑ r = 0 25   25 C r =   50 C 25   × 2 25 = K C 25   50 ∴ K = 2 25

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