Question

A school has 100 students distributed among 1st to 10th standards.
Based on this, which one of the following statements is always correct?

• A. There are at least 10 students who belong to the same standard.
• B. There is at least one student in each standard.
• C. There are at most 10 students in 10th standard.
• D. The total number of students from 1st to 5th standards is at least 50.

Hint:

Think about the pigeonhole principle: if 100 items are placed into 10 boxes, at least one box must hold a certain minimum number of items no matter how they are spread out.

Answer 1

The Correct Option is A

Step 1: Understanding the Question.
There are 100 students split across 10 standards (1st to 10th), but the question does not say how many students are in each standard. We need the ONE statement that must be true for every possible way of splitting 100 students into 10 standards, not just for some special cases.

Step 2: Key Formula or Approach.
Use the pigeonhole principle: if \(n\) items are placed into \(k\) groups, at least one group must contain at least \(\lceil n/k \rceil\) items. Here \(n=100\) students and \(k=10\) standards, so at least one standard must contain at least \(\lceil 100/10 \rceil = 10\) students. This is because if every standard had \(9\) or fewer students, the total would be at most \(9 \times 10 = 90\), which is less than \(100\), a contradiction.

Step 3: Detailed Explanation.
Check option (A): "There are at least 10 students who belong to the same standard." This exactly matches the pigeonhole result above, since at least one standard must hold 10 or more students, no matter how the 100 students are spread out. This is always true.
Check option (B): "There is at least one student in each standard." This fails if, for example, all 100 students are placed in the 1st standard and 0 students in every other standard; the total is still 100, but standards 2 to 10 have no students. So this is not always true.
Check option (C): "There are at most 10 students in 10th standard." This fails if all 100 students are placed in the 10th standard alone; then the 10th standard has 100 students, far more than 10. So this is not always true.
Check option (D): "The total number of students from 1st to 5th standards is at least 50." This fails if all 100 students are placed in the 10th standard; then standards 1st to 5th have 0 students combined, which is less than 50. So this is not always true.

Step 4: Final Answer.
Only option (A) holds for every possible distribution of the 100 students, because the pigeonhole principle guarantees some standard has at least 10 students. \[ \boxed{\text{Option (A)}} \]