Let S = E₁, E₂, , E₁₀ be the sample space of a random experiment such that P(E_n) = n 55 for n = 1, 2, , 10 . If an event A S contains exactly 3 elements and P(A) 2 5 , then the number of such events A is _____.

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Quantrex Academy · Accepted Answer

Correct answer: 16

The probability of event A is the sum of the probabilities of its 3 elements. Given P(A) 2 5 = 22 55 . Let the indices of the 3 elements in A be x, y, z such that x We require x + y + z 22 . The maximum possible sum is 8 + 9 + 10 = 27 . We systematically count the number of valid triplets (x, y, z) for each possible sum: For sum = 27 : 8, 9, 10 ( 1 triplet) For sum = 26 : 7, 9, 10 ( 1 triplet) For sum = 25 : 6, 9, 10 , 7, 8, 10 ( 2 triplets) For sum = 24 : 5, 9, 10 , 6, 8, 10 , 7, 8, 9 ( 3 triplets) For sum = 23 :

Let S = E₁, E₂, , E₁₀ be the sample space of a random experiment such that P(E_n) = n 55 for n = 1, 2, , 10 . If an event A S contains exactly 3 elements and P(A) 2 5 , then the number of such events A is _____.

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