QJEEM21Ma9f98: JEE Main 2021 Mathematics - Class 12 JEEM21Ma9f98 Solution: The probability that a randomly selected 2-digit number belongs …

Question

Verified Solution • 2025–26 Session moderate

The probability that a randomly selected 2-digit number belongs to the set {n

\in

N : (2ⁿ

-

2) is a multiple of 3} is equal to :

Accepted Answer

Total number of cases =

^{90} C_{1} = 90

Now,

2^{n} - 2 = (3 - 1)^{n} - 2

^{n} C_{0} 3^{n} -^{n} C_{1} {.3}^{n - 1} + . . . . + (- 1)^{n - 1} .^{n} C_{n - 1} 3 + (- 1)^{n} .^{n} C_{n} - 2

=

3 (3^{n - 1} - n 3^{n - 2} + . . . + {(- 1)}^{n - 1} . n) + (- 1)^{n} - 2

(2^{n} - 2)

is multiply of 3 only when n is odd

Required Probability

= \frac{45}{90} = \frac{1}{2}

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