The number of students who like their core branches is 1,800.

This GATE CSE 2024 Set 2 question tests Venn diagram concepts for two sets: core branches (C) and other branches (O). Use set theory to solve it step-by-step.

Key Data

• Total students (U) = 10,000
• Students liking neither = 1,500, so |C ∪ O| = 10,000 – 1,500 = 8,500
• |C ∩ O| = 500 (both)
• |C| = (1/4) |O|

Solution Steps

1. Apply the union formula: |C ∪ O| = |C| + |O| – |C ∩ O|
2. Substitute: 8,500 = (1/4)|O| + |O| – 500
3. Simplify: 8,500 + 500 = (5/4)|O|
4. 9,000 = (5/4)|O|
5. |O| = 9,000 × (4/5) = 7,200
6. |C| = 7,200 / 4 = 1,800

Verification

Only core = 1,800 – 500 = 1,300; only other = 7,200 – 500 = 6,700. Total = 1,300 + 6,700 + 500 + 1,500 = 10,000.

Option Analysis

• (A) 1,800: Matches calculation exactly
• (B) 3,500: Too high; assumes |C| = (1/4) of union or ignores both, yielding incorrect |O| = 14,000
• (C) 1,600: Underestimates; possibly from 8,500 – 500 – 3,400 error in ratios
• (D) 1,500: Equals neither; confuses with students liking no branches

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In competitive exams like GATE CSE, engineering college 10000 students core branches problems using Venn diagrams test set theory mastery. This 2024 Set 2 question involves 1,500 students liking neither core nor other branches, core likers as 1/4th of other branch likers, and 500 liking both—find core branch likers.

Why Venn Diagrams Matter

Venn diagrams visualize overlaps: core (C), other (O). Neither gives complement; union formula resolves totals. Common trap: confusing total core with only core.

Full Breakdown

As solved above, |C| = 1,800 fits options perfectly. Practice similar for CSIR NET quantitative aptitude.