Sets, Relations & Functions
Practical Problems on Union and Intersection
Grade 11
Question:
<p>In a class of 140 students numbered 1 to 140, all even numbered students opted Mathematics course, those whose number is divisible by 3 opted Physics course and those whose number is divisible by 5 opted Chemistry course. Then the number of students who did not opt for any of the three courses is:</p>
<p>102</p>
<p>42</p>
<p>1</p>
<p>38</p>
Step-by-Step Solution
Key Concept: Use the inclusion-exclusion principle to find students in at least one course, then subtract from total. Students opting a course have numbers divisible by 2, 3, or 5.
<p><strong>Step 1:</strong> Identify students in each course</p><p>Mathematics (divisible by 2): ⌊140/2⌋ = 70 students</p><p>Physics (divisible by 3): ⌊140/3⌋ = 46 students</p><p>Chemistry (divisible by 5): ⌊140/5⌋ = 28 students</p><p><strong>Step 2:</strong> Find intersections (students in multiple courses)</p><p>Math ∩ Physics (div by 6): ⌊140/6⌋ = 23 students</p><p>Math ∩ Chemistry (div by 10): ⌊140/10⌋ = 14 students</p><p>Physics ∩ Chemistry (div by 15): ⌊140/15⌋ = 9 students</p><p>Math ∩ Physics ∩ Chemistry (div by 30): ⌊140/30⌋ = 4 students</p><p><strong>Step 3:</strong> Apply inclusion-exclusion principle</p><p>|M ∪ P ∪ C| = 70 + 46 + 28 - 23 - 14 - 9 + 4 = 102 students</p><p><strong>Step 4:</strong> Find students who opted for NO course</p><p>Students with no course = 140 - 102 = <strong>38 students</strong></p><p>∴ Answer: D</p>
Correct Answer: D