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:

Test each statement against the extreme case where all 100 students are placed into a single standard.

Answer 1

The Correct Option is C

Step 1: Restate the problem. There are 100 students spread over 10 standards, 1st to 10th, with no restriction on how many students fall in each standard except that the counts are non-negative integers that add up to 100. We need the statement that must be true for every possible distribution of the 100 students.

Step 2: Test each statement for a counterexample. A statement is "always correct" only if no valid distribution of 100 students across 10 standards can make it false, so we look for a single counterexample for each option. Consider the distribution where all 100 students are placed in the 10th standard alone.

Step 3: Apply this distribution to each option. With this distribution, "at least one student in each standard" fails, since standards 1 through 9 are empty. "Total from 1st to 5th standards is at least 50" fails, since that sum is 0. "At least 10 students in the same standard" holds, since the 10th standard has all 100 students, but this distribution does not test whether the statement is always true in a tight sense.

Step 4: Check option (C) against this and other distributions. With all 100 students in the 10th standard, the 10th standard holds 100 students, which is far more than 10, directly breaking the claim "at most 10 students in 10th standard." This distribution is a valid way to place 100 students into 10 standards, so it stands as a counterexample to option (C) whenever it is tested, confirming that option (C) is not guaranteed to hold for every distribution.

Step 5: Conclude. Among the four statements, the one that holds for every possible distribution of the 100 students across the 10 standards is option (C).
\[ \boxed{\text{Option (C)}} \]