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JEE Main202127 Jul 2021Morning ShiftMathematicsProbabilityActual

The probability that a randomly selected 2 - digit number belongs to the set n ∈ N : 2 n - 2 is a multiple of 3 is equal to

Options

  1. A1 6
  2. B2 3
  3. C1 2
  4. D1 3

Correct answer

C. 1 2

Step-by-step solution

Total two-digit numbers is 90 So, total number of cases = C 1 90 = 90 Now, 2 n - 2 = 3 - 1 n - 2 = C 0 n 3 n - C 1 n · 3 n - 1 + … + - 1 n - 1 · C n - 1 n 3 + - 1 n · C n n - 2 = 3 3 n - 1 - n 3 n - 2 + … + - 1 n - 1 · n + - 1 n - 2 Here, the term - 1 n - 2 should be multiple of 3 . So, 2 n - 2 is multiple of 3 only when n is odd. There are total 45 odd two-digit numbers. Hence, the required probability = C 1 45 90 = 45 90 = 1 2

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