Let denote the sum of the numbers obtained when two dice are rolled. If the probability that is , where and are coprime, then find the value of is equal to [2023]

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Solution

Q.
Let $N$ denote the sum of the numbers obtained when two dice are rolled. If the probability that $2^{N} < N!$ is $\frac{m}{n}$ , where $m$ and $n$ are coprime, then $4 m - 3 n$ is equal to [2023]

1 12

2 8

3 6

4 10

Ans.
(2)

$2^{N} < N!, which is true when N \geq 4$

$N = 1 (Not possible)$

$N = 2 i.e., (1, 1) (Not possible)$

$N = 3 i.e., (1, 2), (2, 1) (Not possible)$

$∴$ $Required probability = \frac{36 - 3}{36} = \frac{33}{36} = \frac{11}{12}$

$∴$ $m = 11 and n = 12$

$Now, 4 m - 3 n = 4 (11) - 3 (12) = 44 - 36 = 8$

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