A furniture manufacturer allows customers to customize a dining set by selecting options from four categories:
- Tabletop shape: rectangular, oval, or round (3 choices)
- Wood finish: oak, walnut, cherry, or maple (4 choices)
- Leg design: tapered, hairpin, or turned (3 choices)
- Number of chairs: 4, 6, or 8 (3 choices)
However, due to space constraints, a round tabletop cannot be paired with a set of 8 chairs. How many distinct dining set configurations can a customer assemble?
Cevap: 96
Cevap
96 distinct dining set configurations can be assembled.
To find the number of valid dining set configurations, calculate total unrestricted choices using the Fundamental Counting Principle: multiplying 3 tabletop shapes, 4 wood finishes, 3 leg designs, and 3 chair count options gives 108 total combinations. Next, count the prohibited combinations consisting of a round tabletop (1 option) paired with 8 chairs (1 option) across all 4 finishes and 3 leg designs, giving 1 × 4 × 3 × 1 = 12 restricted combinations. Subtracting 12 restricted combinations from 108 total combinations yields 96 valid configurations.
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Anahtar Kavram
Fundamental Counting Principle with Subtraction of Restricted Cases