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 school has 100 students distributed among 1st to 10th standards.
Based on this, which one of the following statements is always correct?
Based on this, which one of the following statements is always correct?
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Hint:
Try to build a distribution of 100 students across 10 standards that breaks each statement. The one statement you cannot break, no matter how you arrange the students, is the answer.
Try to build a distribution of 100 students across 10 standards that breaks each statement. The one statement you cannot break, no matter how you arrange the students, is the answer.
Updated On: Jul 28, 2026
- There are at least 10 students who belong to the same standard.
- There is at least one student in each standard.
- There are at most 10 students in 10th standard.
- The total number of students from 1st to 5th standards is at least 50.
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The Correct Option is A
Solution and Explanation
Step 1: Understand what "always correct" means here.
The question does not tell us how the 100 students are actually spread across the 10 standards, it could be any distribution. So a statement is "always correct" only if it holds true for every possible way of splitting 100 students into 10 groups, not just for one convenient example.
Step 2: Apply the idea behind the pigeonhole principle.
The pigeonhole principle says that if you place \(n\) items into \(k\) boxes, at least one box must contain at least \(\lceil n/k \rceil\) items, the smallest whole number that is greater than or equal to \(n/k\). Here \(n = 100\) students and \(k = 10\) standards.
Step 3: Compute the guaranteed minimum.
\[ \frac{100}{10} = 10 \]
Since 100 divides evenly by 10, the average number of students per standard is exactly 10. If every standard had strictly fewer than 10 students, the total across all 10 standards would be strictly less than \(10 \times 10 = 100\), which contradicts the fact that there are exactly 100 students in total. So at least one standard must have 10 or more students. This is exactly what option (A) states, so (A) is always true.
Step 4: Check option (B) with a counterexample.
Suppose all 100 students belong to the 1st standard and none belong to standards 2 through 10. This still adds up to 100 students, so it is a valid distribution, but several standards have zero students. So (B) is not always true.
Step 5: Check option (C) with a counterexample.
Suppose all 100 students belong to the 10th standard. This is a valid distribution of 100 students, but it clearly has more than 10 students in the 10th standard. So (C) is not always true.
Step 6: Check option (D) with a counterexample.
Suppose all 100 students belong to standards 6 through 10, with none in standards 1 through 5. The total from 1st to 5th standards is then 0, which is far less than 50. So (D) is not always true.
Final Answer:
Only option (A) is forced to be true no matter how the 100 students are actually distributed among the 10 standards. \[ \boxed{\text{Option (A)}} \]
The question does not tell us how the 100 students are actually spread across the 10 standards, it could be any distribution. So a statement is "always correct" only if it holds true for every possible way of splitting 100 students into 10 groups, not just for one convenient example.
Step 2: Apply the idea behind the pigeonhole principle.
The pigeonhole principle says that if you place \(n\) items into \(k\) boxes, at least one box must contain at least \(\lceil n/k \rceil\) items, the smallest whole number that is greater than or equal to \(n/k\). Here \(n = 100\) students and \(k = 10\) standards.
Step 3: Compute the guaranteed minimum.
\[ \frac{100}{10} = 10 \]
Since 100 divides evenly by 10, the average number of students per standard is exactly 10. If every standard had strictly fewer than 10 students, the total across all 10 standards would be strictly less than \(10 \times 10 = 100\), which contradicts the fact that there are exactly 100 students in total. So at least one standard must have 10 or more students. This is exactly what option (A) states, so (A) is always true.
Step 4: Check option (B) with a counterexample.
Suppose all 100 students belong to the 1st standard and none belong to standards 2 through 10. This still adds up to 100 students, so it is a valid distribution, but several standards have zero students. So (B) is not always true.
Step 5: Check option (C) with a counterexample.
Suppose all 100 students belong to the 10th standard. This is a valid distribution of 100 students, but it clearly has more than 10 students in the 10th standard. So (C) is not always true.
Step 6: Check option (D) with a counterexample.
Suppose all 100 students belong to standards 6 through 10, with none in standards 1 through 5. The total from 1st to 5th standards is then 0, which is far less than 50. So (D) is not always true.
Final Answer:
Only option (A) is forced to be true no matter how the 100 students are actually distributed among the 10 standards. \[ \boxed{\text{Option (A)}} \]
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