All of the students in a certain class took a test consisting of 20

Question

All of the students in a certain class took a test consisting of 20 questions. Did at least two students have the same number of correct answers on the test?

(1) Each student answered an even number of questions correctly.
(2) There are more than 11 students in the class.

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ManifestDreamMBA · Answer

Total Q = 20
Total Students = S

Statement 1
Each student answered even number of questions correctly.
We have 11 possible values 0,2,4,....,20
If S> 11, then answer is Yes else No
Without knowing the number of students, we cannot say if 2 students have the same number of correct answers
Insufficient

Statement 2
S >11
The possible number of correct answers for each of them could be anything
1,2,3,4,5,...,11 OR All 20
Insufficient

Combined, we know the answer is Yes as S > 11 and this was the missing piece in Statement 1 to answer the prompt. Answer C

GMATinsight · Answer

Question: Did atleast two students have the same number of correct answers?

Given: Total Questions = 20

Statement 1: Each student answered an even number of questions correctly.
i.e. Correct answers of students are {2, 4, 6, 8, 10, 12, 14, 16, 18, 20} [10 possibilities]
But the number of students is unknown where as we know that there could be only 10 possible different response outcomes as mentioned above hence
NOT SUFFICIENT

Statement 2: There are more than 11 students in the class.
For 12 students their correct number of questions could be {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}i.e. NO to question
For 12 students their correct number of questions could be {1, 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} i.e. YES to question
NOT SUFFICIENT

Combining the statements
As per statement-1, there can be only 10 possible outcomes of number of correct
Since the number of students is greater than 11 (statetemtn2) therefore atleast two students MUST have same number of correct hence
SUFFICIENT

Answer: Option C

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Bunuel

All of the students in a certain class took a test consisting of 20 questions. Did at least two students have the same number of correct answers on the test?

(1) Each student answered an even number of questions correctly.
(2) There are more than 11 students in the class.

NextstopISB · Answer

We need to determine whether at least two students had the same number of correct answers on a 20-question test.

Step 1: Understanding the Problem 
Each student’s score is a whole number between 0 and 20. We need to check if any two students had the same score. 
This is a pigeonhole principle question: If we have more students than possible distinct scores, at least two must share the same score.

Step 2: Evaluating Statement (1) 
(1) Each student answered an even number of questions correctly. 
- Possible scores are limited to even numbers: {0, 2, 4, ..., 20}. 
- There are 11 possible scores.

However, we don’t know how many students are in the class. If there are more than 11 students, the pigeonhole principle would guarantee a duplicate score. If there are 11 or fewer, we cannot be sure. 
Insufficient.

Step 3: Evaluating Statement (2) 
(2) There are more than 11 students in the class. 
- The number of students is at least 12. 
- However, this doesn’t tell us about the possible scores they achieved. If all 21 scores (0 to 20) were available, we could have 12 different scores without duplication. 
Insufficient.

Step 4: Evaluating Statements Together 
- From (1), possible scores are only 11 different even numbers. 
- From (2), there are more than 11 students. 
- Since there are more students than available distinct scores, at least two must share the same score.

Together, they are sufficient.

Final Answer: (C) Together sufficient, but neither alone.

GMATinsight · Answer

Question: Did two students have same number of correct answers?

Statement 1  : Each student answered an even number of questions correctly.

i.e. students answered {0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20} - 11 total choices
but we don't know how many students the class has therefore
 NOT SUFFICIENT

Statement 2  : There are more than 11 students in the class.
But we don't know scoring method hence 
 NOT SUFFICIENT

Combining the statements  
more than 11 students and only 11 possible outcomes of correct answers hence definitely atleast two marked the same number of correct answers hence
 SUFFICIENT

Answer: Option C

Related Video:  https://www.youtube.com/watch?v=ZTQzhgqxZ1Y

All of the students in a certain class took a test consisting of 20

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All of the students in a certain class took a test consisting of 20

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