Question

Two integers \(r\) and \(s\) are drawn one at a time without replacement from the set \(\{1, 2, \ldots, n\}\). Then \(P(r \leq k / s \leq k)\) is:

• A. \(\frac{k}{n}\)
• B. \(\frac{k}{n-1}\)
• C. \(\frac{k-1}{n}\)
• D. \(\frac{k-1}{n-1}\)

Hint:

When selecting from a set without replacement, consider the total number of ways to select and the number of favorable outcomes, then calculate the probability.

Answer 1

The Correct Option is D

Step 1: The problem asks for the probability that \(r \leq s \leq k\), where \(r\) and \(s\) are chosen from the set \(\{1, 2, \dots, n\}\).

Step 2: First, count the total number of ways to choose two integers from \(\{1, 2, \dots, n\}\). This is \(\binom{n}{2}\).

Step 3: Now, count the favorable outcomes where \(r \leq s \leq k\). The number of such pairs is \(k - 1\) because the integers \(r\) and \(s\) must be less than or equal to \(k\) and ordered accordingly.

Step 4: The probability is the ratio of favorable outcomes to total outcomes, which simplifies to \(\frac{k-1}{n-1}\).