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Read the below information carefully and answer the following questions.
There are 100 students in a class. Each of the students has to opt for one or more of the three specializations among Finance, Operations and Marketing. The number of students who opted for Marketing is more than the number of students who opted for Finance which is more than the number of students who opted for Operations which is more than the number of students who opted for exactly two of the three specializations which is more than the number of students who opted for all three specializations. At least one student opted for all three specializations.
 
Q. What is the maximum number of students who opted for Operations as a specialization?
    Correct answer is '82'. Can you explain this answer?
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    Read the below information carefully and answer the following question...
    It is known that the total number of students is 100. We also know that the number of students who opted for Marketing is more than the number of students who opted for Finance which is more than the number of students who opted for Operations.
    To maximize the number of students who opted for Operations we have to maximize the number of students who study two specializations and three specializations.
    This can be done by assigning 49 students to those who have opted for all three specializations and 51 students to those who have opted exactly two specializations. The 51 students have to be divided among Marketing, Finance and Operations as equally as possible which is as shown in the Venn diagram below.

    Thus, the maximum number of students who opted for Operations as a specialization = 16+17+49 = 82
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    Read the below information carefully and answer the following question...
    **Understanding the given information:**

    - There are 100 students in the class.
    - Each student can opt for one or more of the three specializations: Finance, Operations, and Marketing.
    - The number of students who opted for Marketing is more than the number of students who opted for Finance.
    - The number of students who opted for Finance is more than the number of students who opted for Operations.
    - The number of students who opted for Operations is more than the number of students who opted for exactly two specializations.
    - The number of students who opted for exactly two specializations is more than the number of students who opted for all three specializations.
    - At least one student opted for all three specializations.

    **Finding the maximum number of students who opted for Operations:**

    To find the maximum number of students who opted for Operations, we need to analyze the given information and make logical deductions.

    Let's assume the number of students who opted for Operations as x.

    1. The number of students who opted for exactly two specializations is more than the number of students who opted for all three specializations.
    - This implies that the number of students who opted for exactly two specializations is greater than 0.
    - Therefore, the number of students who opted for Operations (x) is greater than the number of students who opted for all three specializations (0).

    2. The number of students who opted for Operations is more than the number of students who opted for exactly two specializations.
    - This means that x is greater than the number of students who opted for exactly two specializations.

    3. The number of students who opted for Finance is more than the number of students who opted for Operations.
    - This implies that the number of students who opted for Operations (x) is less than the number of students who opted for Finance.

    4. The number of students who opted for Marketing is more than the number of students who opted for Finance.
    - This means that the number of students who opted for Finance is less than the number of students who opted for Marketing.

    Based on the above deductions, we can conclude the following:

    0 < all="" three="" specializations="" />< exactly="" two="" specializations="" />< operations="" />< finance="" />< />

    Since the total number of students is 100, and at least one student opted for all three specializations, the maximum number of students who opted for Operations (x) can be calculated as follows:

    All three specializations + Exactly two specializations + Operations + Finance + Marketing = 100

    Let's substitute the inequalities into the equation:

    0 + Exactly two specializations + x + (x + 1) + (x + 2) = 100

    Simplifying the equation:

    3x + 3 = 100

    3x = 97

    x = 32.33

    Since the number of students must be a whole number, the maximum number of students who opted for Operations is 32. This means that the correct answer is 32 and not 82 as mentioned in the question.
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