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Sum N of two dice: probability that N satisfies a condition

This type of question, asked 2 times: 25 Jan 2023, 10 Apr 2023. All 2 questions below, oldest first.

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  1. Question 1

    25 Jan 2023

    162
    Let NN be the sum of the numbers appeared when two fair dice are rolled and let the probability that N−2,3 N, N+2N - 2,\sqrt{3\text{ }N},\text{ }N + 2 are in geometric progression be k48\frac{k}{48}. Then the value of kk is

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  2. Question 2

    10 Apr 2023

    72
    Let NN denotes the sum of the numbers obtained when two dice are rolled. If the probability that 2N<N2^{N}< N ! is mn\frac{m}{n}, where mm and nn are coprime, then 4m−3n4m - 3n is equal to

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