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Difficulty: MediumFundamental Counting Principle

An architectural firm offers custom townhouse design packages created by making sequential choices across three stages: Exterior Facade Material (4 choices), Floor Plan Layout (5 choices), and Interior Lighting Scheme (6 choices).

However, two design constraints apply:
1. If the first exterior facade material option is chosen, the fifth floor plan layout option cannot be selected.
2. The sixth interior lighting scheme option can only be selected if the fifth floor plan layout option is chosen.

How many different valid townhouse design packages can a client create?

Answer: 98 packages

Answer

The total number of valid townhouse design packages is 98.
The solution splits the decision process into two mutually exclusive scenarios based on whether the 5th layout option is chosen. In Case 1 (5th layout chosen), 3 facade choices × 1 layout choice × 6 lighting choices yields 18 valid packages. In Case 2 (1st through 4th layout chosen), 4 facade choices × 4 layout choices × 5 lighting choices yields 80 valid packages. Adding these mutually exclusive totals gives 18 + 80 = 98 valid design packages.

Step-by-Step Solution

1
Analyze the decision tree by splitting into mutually exclusive cases based on whether Floor Plan Layout Option 5 is selected.
Two distinct cases are established: Case 1 (Layout Option 5 selected) and Case 2 (Layout Options 1–4 selected).
The constraints depend directly on whether Floor Plan Layout Option 5 is chosen.
2
Calculate valid combinations for Case 1 (Layout Option 5 selected).
Facade choices = 3 (excluding Option 1), Layout choices = 1 (Option 5), Lighting choices = 6 (including Option 6). Total = 3 × 1 × 6 = 18 packages.
Constraint 1 eliminates 1 facade option, while Constraint 2 allows all 6 lighting options.
3
Calculate valid combinations for Case 2 (Layout Option 5 NOT selected).
Facade choices = 4, Layout choices = 4 (Options 1–4), Lighting choices = 5 (excluding Option 6). Total = 4 × 4 × 5 = 80 packages.
Constraint 1 does not apply, but Constraint 2 prevents selecting Lighting Option 6.
4
Sum the valid outcomes from both mutually exclusive cases.
18 + 80 = 98 valid packages.
By the Addition Principle of Counting, non-overlapping cases are added together.

Key Concept

Fundamental Counting Principle with Constrained Cases
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Topics

Fundamental Counting Principle

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