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Difficulty: MediumOverlapping Sets and Inclusion-Exclusion Principle

A research group evaluated 150 renewable energy projects. Each project utilizes at least one of three primary energy technologies: solar, wind, or hydro. Exactly 85 projects utilize solar power, 60 projects utilize wind power, and 45 projects utilize hydro power. Furthermore, 20 projects utilize both solar and wind power, 15 projects utilize both solar and hydro power, and 10 projects utilize both wind and hydro power. How many of the surveyed projects utilize all three energy technologies?

Answer: 5 projects

Answer

5 projects utilize all three energy technologies.
Applying the three-set inclusion-exclusion principle Total = |A| + |B| + |C| - (|A ∩ B| + |A ∩ C| + |B ∩ C|) + |A ∩ B ∩ C| + Neither gives 150 = 85 + 60 + 45 - (20 + 15 + 10) + |A ∩ B ∩ C| + 0. Simplifying yields 150 = 145 + |A ∩ B ∩ C|, which gives |A ∩ B ∩ C| = 5.

Step-by-Step Solution

1
Identify known set sizes and recall the Principle of Inclusion-Exclusion for three sets.
Total = 150, |S| = 85, |W| = 60, |H| = 45, |S ∩ W| = 20, |S ∩ H| = 15, |W ∩ H| = 10, Neither = 0.
Setting up known quantities ensures proper application of set intersection rules.
2
Substitute the set sizes into the formula: Total = |S| + |W| + |H| - (|S ∩ W| + |S ∩ H| + |W ∩ H|) + |S ∩ W ∩ H| + Neither.
150 = 85 + 60 + 45 - (20 + 15 + 10) + |S ∩ W ∩ H| + 0.
Pairwise overlaps are double-counted when individual sets are summed and must be subtracted; the central triple overlap is over-subtracted and must be added back.
3
Combine known terms and isolate the unknown triple intersection.
150 = 190 - 45 + |S ∩ W ∩ H| => 150 = 145 + |S ∩ W ∩ H| => |S ∩ W ∩ H| = 5.
Subtracting 145 from 150 yields the exact count of projects belonging to all three sets.

Key Concept

Three-Set Inclusion-Exclusion Principle
Estimated Time:2m 0s
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Topics

Overlapping Sets and Inclusion-Exclusion Principle

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