Sets, Relations & Functions
Sets
Grade None
Question:
<p>In a group of 140 students, 70 opted Mathematics, 46 opted Physics and 28 opted Chemistry. 23 opted both Mathematics and Physics, 9 opted both Physics and Chemistry, 14 opted both Mathematics and Chemistry, and 4 opted all three subjects. The number of students who did not opt for any of the three courses is:</p>
<p>20</p>
<p>28</p>
<p>32</p>
<p>38</p>
Step-by-Step Solution
Key Concept: Use the inclusion-exclusion principle: |M ∪ P ∪ C| = |M| + |P| + |C| - |M ∩ P| - |P ∩ C| - |M ∩ C| + |M ∩ P ∩ C|. Students opting no course = Total - |M ∪ P ∪ C|.
<p><strong>Step 1:</strong> Apply inclusion-exclusion principle for three sets.</p><p>|M ∪ P ∪ C| = |M| + |P| + |C| - |M ∩ P| - |P ∩ C| - |M ∩ C| + |M ∩ P ∩ C|</p><p><strong>Step 2:</strong> Substitute the given values:</p><p>|M ∪ P ∪ C| = 70 + 46 + 28 - 23 - 9 - 14 + 4</p><p>|M ∪ P ∪ C| = 144 - 46 + 4</p><p>|M ∪ P ∪ C| = 102</p><p><strong>Step 3:</strong> Find students who opted for no courses:</p><p>Students with no courses = Total - |M ∪ P ∪ C|</p><p>Students with no courses = 140 - 102 = 38</p><p>∴ Answer: D (38)</p>
Correct Answer: D