Correct : a

Explanation:
Let us solve this problem using a Venn diagram approach.
1. Identify the Total Population Breakdown:
• Total number of students = 10,000
• Students who like neither branch = 1,500
• Therefore, the total number of students who like at least one branch (the union of both sets) is:
    10,000 - 1,500 = 8,500
2. Define the Variables:
• Let C be the total number of students who like their core branches.
• Let O be the total number of students who like other branches.
• The problem states that the number of students who like both branches is:
    C ∩ O = 500
3. Set up the Algebraic Equations:
• According to the problem, the number of students who like core branches is 1/4th of those who like other branches:
    C = O / 4 → O = 4C
• Using the standard set union formula:
    Total Union = C + O - (C ∩ O)
    8,500 = C + 4C - 500
4. Solve for C:
    8,500 + 500 = 5C
    9,000 = 5C
    C = 9,000 / 5 = 1,800