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: 100 students placed into 10 standards forces at least one standard to have 100/10 = 10 or more students.

Answer 1

The Correct Option is A

Step 1: Understand what "always correct" means.
The 100 students can be split among the 10 standards in any way at all, some standards could have very few students and others could have a lot, as long as the total is 100. A statement is "always correct" only if it stays true for every single one of these possible splits, not just for one convenient example.

Step 2: Apply the pigeonhole principle to test option (A).
Suppose, for the sake of argument, that no standard has 10 or more students, so every standard has at most 9 students. There are 10 standards, so the largest possible total under this assumption is
\[ 9 \times 10 = 90 \]
This is less than 100, which contradicts the fact that there are 100 students in total. So the assumption "every standard has at most 9 students" cannot be true. This means at least one standard must have 10 or more students, which is exactly what option (A) says. This is the pigeonhole principle: 100 students placed into 10 groups force at least one group to hold at least \(\lceil 100/10 \rceil = 10\) students.

Step 3: Show the other options can fail, using specific distributions.
For option (B), put all 100 students in the 1st standard and 0 in every other standard. This is a valid distribution of 100 students among 10 standards, but standards 2nd to 10th have zero students, so "at least one student in each standard" is false here. Option (B) is not always true.
For option (C), put all 100 students in the 10th standard. Then the 10th standard has 100 students, far more than 10, so "at most 10 students in 10th standard" is false. Option (C) is not always true.
For option (D), put all 100 students in standards 6th to 10th (for example, 20 in each of those five standards) and 0 in standards 1st to 5th. Then the total from 1st to 5th standards is 0, which is less than 50, so "at least 50 students from 1st to 5th" is false. Option (D) is not always true.

Step 4: Final Answer.
Only option (A) holds for every possible distribution of the 100 students, because it follows directly from the pigeonhole principle. \[ \boxed{\text{Option (A)}} \]