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JEE Main202126 Aug 2021Evening ShiftMathematicsBinomial TheoremActual

Let n k denote C k n and n k = n k , if 0 ≤ k ≤ n 0 , otherwise . If A k = ∑ i = 0 9 9 i 12 12 - k + i + ∑ i = 0 8 8 i 13 13 - k + i and A 4 - A 3 = 190 p , then p is equal to _______.

Correct answer

0

Step-by-step solution

A k = ∑ i = 0 9 9 C i · C k - i 12 + ∑ i = 0 8 8 C i · C k - i 13 = Coefficient of x k in 1 + x 9 × 1 + x 12 + Coefficient of x k in 1 + x 8 × 1 + x 13 = Coefficient of x k in 1 + x 21 + Coefficient of x k in 1 + x 21 A k = C k 21 + C k 21 = 2 . C k 21 A 4 = ∑ i = 0 9 9 C i · C 4 - i 12 + ∑ i = 0 8 8 C i · C 4 - i 13 = C 4 21 + C 4 21 = 2 C 4 21 A 3 = C 3 21 + C 3 21 = 2 C 3 21 A 4 - A 3 = 2 C 4 21 - C 3 21 = 190 p ⇒ p = 49

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