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: Count by choosing seats:
Number the seats 1 to 9. Girls must take seat 1 and seat 9, and one more seat from 3 to 7 (seat 2 and 8 are next to the end girls, so they are not allowed). That gives 5 choices for the third seat.

Step 2: Fill the seats:
The 3 girls can be placed on the chosen seats in $3! = 6$ ways, and the boys fill the other 6 seats in $6! = 720$ ways.
Favourable = $5\times 6\times 720 = 21600$. Total = $9! = 362880$.
\[ P = \frac{21600}{362880} = \frac{5}{84} \]
This is option (D).

Final Answer:
$\frac{5}{84}$. \[ \boxed{\frac{5}{84}} \]