Question

An institute has 5 departments and each department has 50 students. If students are picked up randomly from all 5 departments to form a committee, what should be the minimum number of students in the committee so that at least one department should have representation of minimum 5 students?

• A. 11
• B. 15
• C. 21
• D. 41
• E. None of the above.

Hint:

When a question asks for the minimum number to guarantee a certain count in at least one group, think “worst-case distribution” and apply the Pigeonhole Principle.

Answer 1

The Correct Option is C

Step 1: Use the Pigeonhole Principle (worst-case packing).
To \emph{avoid} having 5 students from any one department for as long as possible, pick at most 4 from each department first. With 5 departments, the maximum students you can pick without hitting 5 from any department is: \[ 4 \times 5 = 20. \]

Step 2: Force the threshold.
The next pick (the \(21^{\text{st}}\) student), no matter which department they belong to, will raise some department’s count from 4 to 5. Hence, 21 students are sufficient (and 20 are not).

Final Answer: \[ \boxed{\text{C. 21}} \]