There are 20 questions in a question paper. If no two students solve the same combination of questions but solve equal number of questions then the maximum number of students who appeared in the examination isa)20C9b)20C11c)20C10d)none of theseCorrect answer is option 'C'. Can you explain this answer?

Question

Answer

Explanation:

To find the maximum number of students who appeared in the examination, we need to determine the number of different combinations of questions that can be solved by each student.

Given:
- There are 20 questions in the question paper.
- No two students solve the same combination of questions.
- All students solve an equal number of questions.

Step 1: Determine the number of questions solved by each student
Since all students solve an equal number of questions, we need to find a factor of 20 that will give us the number of questions solved by each student.

Factors of 20 are: 1, 2, 4, 5, 10, 20.

Step 2: Calculate the number of combinations
For each factor, we can calculate the number of combinations of questions that can be solved by the students.

For example, if a student solves 1 question, there are 20C1 = 20 ways to choose the question.

Similarly, if a student solves 2 questions, there are 20C2 = 190 ways to choose the questions.

Calculating the combinations for each factor gives us the following:

1 question: 20C1 = 20
2 questions: 20C2 = 190
4 questions: 20C4 = 4845
5 questions: 20C5 = 15504
10 questions: 20C10 = 184,756
20 questions: 20C20 = 1

Step 3: Determine the maximum number of students
To find the maximum number of students, we need to find the factor that results in the maximum number of combinations.

From the calculations above, we can see that the maximum number of combinations is obtained when a student solves 10 questions (20C10 = 184,756).

Therefore, the maximum number of students who appeared in the examination is 184,756.

Conclusion:
The correct answer is option C (20C10).

Answer

If r questions are solved by each student then the number of possible selections of questions is ²⁰C_r
∴ the number of students =²⁰C_r
(∴ each student has solved different combination of questions)
∴ the maximum number of students = maximum value of ²⁰C_r = ²⁰C₁₀, because ²⁰C₁₀ is the largest among ²⁰C₀,²⁰C₁, ....²⁰C₂₀ - being the middle one.

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