Question

A school has 100 students distributed among 1^st to 10^th 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 10^th standard.
• D. The total number of students from 1^st to 5^th standards is at least 50.

Hint:

Use the pigeonhole principle: with 100 students in 10 standards, if every standard had 9 or fewer, the total could not reach 100.

Answer 1

The Correct Option is A

A quick way to see this is through the idea of an average. If 100 students are spread over 10 standards, the average number of students per standard is $100/10=10$. Whenever a set of whole numbers has an average of 10, at least one of them must be 10 or more, since it is impossible for every single value to sit strictly below the average.

1. There are at least 10 students who belong to the same standard: matches the averaging argument directly, since some standard must reach or cross the average of 10. This is always true.
2. There is at least one student in each standard: the 100 students could all be crammed into one standard, leaving the other 9 empty, so this can be false.
3. There are at most 10 students in 10th standard: the 10th standard alone could hold anywhere from 0 to 100 students, so a cap of 10 is not forced.
4. The total from 1st to 5th standards is at least 50: all 100 students could be pushed into standards 6 to 10, making the 1st to 5th total as low as 0.

Every option except the first can be broken by some valid distribution of the 100 students, which means only the first statement is guaranteed under every possible distribution.

Let's summarize:

• Average students per standard is 10, so some standard must have 10 or more.
• Every other option can be defeated by an extreme, uneven distribution of students.

The statement that always holds is "there are at least 10 students who belong to the same standard", option (A).