16 students were writing a test in a class. Rahul made 14 mistakes in the paper, which was the highest number of mistakes made by any student. Which of the following statements is definitely true?

Options

A. At least two students made the
same number of mistakes.
B. Exactly two students made the
same number of mistakes.
C. At most two students made the
same number of mistakes.
D. All students made different
number of mistakes.

Answer

To solve the problem step by step, we need to analyze the information given and draw conclusions based on it.

### Step 1: Understand the Problem
We know that there are 16 students in total, and Rahul made 14 mistakes, which is the highest number of mistakes made by any student.

**Hint:** Identify the key numbers in the problem: total students (16) and highest mistakes (14).

### Step 2: Determine the Range of Mistakes
Since Rahul made 14 mistakes, the number of mistakes made by the other students must be less than or equal to 14. The minimum number of mistakes a student can make is 0. Therefore, the range of mistakes made by the students is between 0 and 14.

**Hint:** Establish the minimum and maximum limits for the number of mistakes.

### Step 3: Analyze the Number of Unique Mistake Counts
The possible unique counts of mistakes that can be made by the students are the integers from 0 to 14. This gives us a total of 15 possible distinct values (0, 1, 2, ..., 14).

**Hint:** Count the distinct values from the minimum to the maximum.

### Step 4: Apply the Pigeonhole Principle
Since there are 16 students and only 15 distinct possible values for the number of mistakes, the Pigeonhole Principle states that if you have more items (students) than containers (distinct mistake counts), at least one container must contain more than one item. This means at least two students must have made the same number of mistakes.

**Hint:** Recall the Pigeonhole Principle which states that if n items are put into m containers, with n > m, then at least one container must contain more than one item.

### Step 5: Conclusion
From the analysis, we can conclude that at least two students must have made the same number of mistakes. Therefore, the statement that is definitely true is that at least two students made the same number of mistakes.

**Hint:** Look for statements that can be logically concluded from the given information.

### Final Answer
At least two students made the same number of mistakes.