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?
Show Hint
Use the pigeonhole principle: with 100 students in 10 standards, if every standard had 9 or fewer, the total could not reach 100.
Updated On: Jul 20, 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.
Show Solution
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The Correct Option is A
Solution and Explanation
Step 1: Identify the concept.
This is a pigeonhole principle question. There are 100 students (the "pigeons") placed into 10 standards (the "pigeonholes"), and we need the one statement that is forced to be true no matter how the 100 students are spread across the 10 standards.
Step 2: Test option (A) using the pigeonhole principle.
Suppose, for contradiction, that every standard has at most 9 students. Then the total number of students across all 10 standards can be at most \(10 \times 9 = 90\).
But we are told there are 100 students, and \(100 > 90\), which is a contradiction.
So at least one standard must contain 10 or more students. This means "there are at least 10 students who belong to the same standard" is always true.
Step 3: Check why option (B) can fail.
It is possible to put all 100 students into just the 1st standard and leave every other standard empty. Then some standards have zero students, so "at least one student in each standard" is not guaranteed.
Step 4: Check why option (C) can fail.
Nothing stops all 100 students from being placed in the 10th standard alone. Then the 10th standard has 100 students, which is far more than 10, so "at most 10 students in 10th standard" is not guaranteed.
Step 5: Check why option (D) can fail.
Place all 100 students in standards 6 to 10 only, with none in standards 1 to 5. Then the total from 1st to 5th standards is 0, which is less than 50, so this statement is not guaranteed either.
Final Answer:
Only the pigeonhole-based statement always holds. \[ \boxed{\text{Option (A)}} \]
This is a pigeonhole principle question. There are 100 students (the "pigeons") placed into 10 standards (the "pigeonholes"), and we need the one statement that is forced to be true no matter how the 100 students are spread across the 10 standards.
Step 2: Test option (A) using the pigeonhole principle.
Suppose, for contradiction, that every standard has at most 9 students. Then the total number of students across all 10 standards can be at most \(10 \times 9 = 90\).
But we are told there are 100 students, and \(100 > 90\), which is a contradiction.
So at least one standard must contain 10 or more students. This means "there are at least 10 students who belong to the same standard" is always true.
Step 3: Check why option (B) can fail.
It is possible to put all 100 students into just the 1st standard and leave every other standard empty. Then some standards have zero students, so "at least one student in each standard" is not guaranteed.
Step 4: Check why option (C) can fail.
Nothing stops all 100 students from being placed in the 10th standard alone. Then the 10th standard has 100 students, which is far more than 10, so "at most 10 students in 10th standard" is not guaranteed.
Step 5: Check why option (D) can fail.
Place all 100 students in standards 6 to 10 only, with none in standards 1 to 5. Then the total from 1st to 5th standards is 0, which is less than 50, so this statement is not guaranteed either.
Final Answer:
Only the pigeonhole-based statement always holds. \[ \boxed{\text{Option (A)}} \]
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