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: 100 students split into 10 standards gives an average of 10 students per standard.

Answer 1

The Correct Option is A

Step 1: Identify the setup.
There are 100 students split among exactly 10 standards (1st to 10th), and the question does not say the students are spread evenly, so any distribution of positive whole numbers that adds up to 100 across the 10 standards is possible.

Step 2: Recall the pigeonhole principle.
The pigeonhole principle states that if \(n\) items are placed into \(k\) groups, then at least one group must contain at least \(\lceil n/k \rceil\) items (the ceiling of \(n\) divided by \(k\)). Here \(n = 100\) students and \(k = 10\) standards, so at least one standard must contain at least \(\lceil 100/10 \rceil = 10\) students.

Step 3: Confirm option (A) is always true.
Since \(100/10 = 10\) exactly, it is impossible for every standard to have fewer than 10 students, because 10 standards with at most 9 students each would total at most \(10 \times 9 = 90\), which is less than 100. So at least one standard must have 10 or more students, which is exactly what option (A) states.

Step 4: Rule out option (B).
Option (B) says every standard has at least one student. This is not guaranteed, since all 100 students could be placed only in the 1st standard, leaving the other 9 standards empty, so option (B) can be false.

Step 5: Rule out option (C).
Option (C) says at most 10 students are in the 10th standard specifically. This is not guaranteed either, because all 100 students could be placed in the 10th standard alone, so option (C) can be false.

Step 6: Rule out option (D).
Option (D) says the total from 1st to 5th standards is at least 50. This can be false too, for instance if all 100 students are placed in the 6th to 10th standards, then the 1st to 5th standards would have 0 students, well below 50.

\[ \boxed{\text{Option (A) is always correct}} \]