Question

In a test, there are 5 questions of the 'true or false' type. No student has got all the answers correct and the sequence of answers for every student is unique. The maximum number of students who could have appeared for the test is....

• A. \(5\)
• B. \(31\)
• C. \(15\)
• D. \(89\)

Hint:

Seat the boys first, then place girls in the gaps between boys.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Total seating arrangements of 9 people in a row: \(9!\). We count the arrangements where the two end seats hold girls and no two girls are adjacent.

Step 2: Arrange the boys first:
The 6 boys can be arranged in \(6!\) ways. They create 7 gaps, including the 2 end gaps. Girls placed in separate gaps are never adjacent.

Step 3: Place the girls:
Two girls must sit in the two end gaps and the third girl in one of the 5 inner gaps (5 ways). The three girls can be assigned to these three gaps in \(3!\) ways.
Favourable arrangements = \(6!\times 5\times 3!\).

Step 4: Probability:
\[ P = \frac{6!\times 5\times 3!}{9!} = \frac{5\times 6}{9\times 8\times 7} = \frac{30}{504} = \frac{5}{84} \]
A girl at each end with the third girl in an inner gap automatically means no two girls are adjacent, because each is separated by at least one boy.

Final Answer:
The required probability is \(\frac{5}{84}\), option (D). \[ \boxed{\frac{5}{84}} \]