Multiple choice

A bag contains 30 tickets, numbered 1 to 30. Five tickets are drawn at random and arranged in ascending order. Find the probability that the 3rd number is 20.

  1. 285/5278

  2. 185/5278

  3. 291/3452

  4. 192/3455
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A Correct answer
Explanation

Total ways to choose 5 tickets from 30 is 30C5. For the 3rd number to be 20, we must choose 2 numbers from {1, ..., 19} and 2 numbers from {21, ..., 30}. Ways = 19C2 * 10C2 = (19*18/2) * (10*9/2) = 171 * 45 = 7695. Total ways = 30C5 = 142506. Probability = 7695 / 142506 = 285 / 5278.

AI explanation

The total number of ways to draw five tickets from thirty is 30C5 = 142506. For the third ticket in the ascending sequence to be 20, exactly two of the drawn tickets must be from the numbers 1 through 19, and exactly two must be from the numbers 21 through 30. The number of successful combinations is 19C2 * 10C2 = 171 * 45 = 7695. The probability is 7695/142506, which simplifies by dividing the numerator and denominator by 27 to 285/5278.

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