Question

Two numbers are selected at random without replacement from the first \(n\) natural numbers. Let X denote the larger of the two numbers obtained. If \(E(X)=\frac{14}{3}\), then the value of \(n\) is

• A. \(2\)
• B. \(4\)
• C. \(6\)
• D. \(8\)

Hint:

Find the probability that the larger number is k, form the expectation, and use the sum of k(k - 1).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Choose 2 numbers from \(1,2,\dots,n\). The total number of selections is \(\binom{n}{2}\).

Step 2: Probability Distribution
\(X=k\) (for \(k=2,\dots,n\)) when the larger number is \(k\) and the other is any of \(k-1\) smaller numbers.
\(P(X=k)=\dfrac{k-1}{\binom{n}{2}}\).

Step 3: Find E(X)
\(E(X)=\dfrac{\sum_{k=1}^{n}k(k-1)}{\binom{n}{2}}=\dfrac{\frac{(n-1)n(n+1)}{3}}{\frac{n(n-1)}{2}}=\dfrac{2(n+1)}{3}\).

Step 4: Solve
\(\dfrac{2(n+1)}{3}=\dfrac{14}{3}\Rightarrow n+1=7\Rightarrow n=6\).

Final Answer:
The value of \(n\) is 6. \[ \boxed{\text{(C) }6} \]