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JEE Main201910 Jan 2019Morning ShiftMathematicsBinomial TheoremActual

If ∑ i = 1 20 20 C i - 1 20 C i + 20 C i - 1 3 = k 21 , then k equals

Options

  1. A200
  2. B100
  3. C50
  4. D400

Correct answer

B. 100

Step-by-step solution

Given ∑ i = 1 20   20 C i - 1   20 C i +   20 C i - 1 3 = k 21 We have   20 C i - 1   20 C i +   20 C i - 1 Using C r - 1 n + C r n = C r n + 1 , we get   20 C i - 1   20 C i +   20 C i - 1 =   20 C i - 1   21 C i Now, using C r n = n r × C r - 1 n - 1 , we get   20 C i - 1   21 C i = 20 C i - 1 21 i × 20 C i - 1 = i 21 Hence, the given sum is ∑ i = 1 20 i 3 21 3 = 1 21 3 ∑ i = 1 20 i 3 Now, using ∑ r 3 r = 1 n = n n

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