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JEE Main202126 Feb 2021Morning ShiftMathematicsBinomial TheoremActual

Let m , n ∈ N and gcd 2 , n = 1 . If 30 30 0 + 29 30 1 + … . . + 2 30 28 + 1 30 29 = n . 2 m , then n + m is equal to _______. (Here n k = C k n )

Correct answer

0

Step-by-step solution

30 C 0 30 + 29 C 1 30 + … + 2 C 28 30 + 1 C 29 30 = 30 C 30 30 + 29 C 29 30 + … … + 2 C 2 30 + 1 C 1 30 = ∑ r = 1 30 r C r 30 = ∑ r = 1 30 r 30 r C r - 1 29 = 30 ∑ r = 1 30 C r - 1 29 = 30 C 0 29 + C 1 29 + C 2 29 + … + C 29 29 = 30 2 29 = 15 2 30 = n 2 m ∴   n = 15 , m = 30 n + m = 45

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