Approach

To effectively answer the question about writing a function to determine the number of ways to paint houses using k different colors, follow this structured framework:

1. Understand the Problem: Clearly define the problem by identifying the constraints and requirements.

2. Identify the Inputs and Outputs: Determine what inputs your function will take and what output it will produce.

3. Develop a Strategy: Choose an appropriate algorithm or mathematical approach to solve the problem.

4. Implement the Function: Write the code for the function, ensuring clarity and efficiency.

5. Test the Function: Validate your function with various test cases to ensure it works as expected.

Key Points

• Understanding Constraints: Know if there are any restrictions on how colors can be applied to adjacent houses.

• Dynamic Programming: This is often the most effective way to approach problems involving combinations and permutations.

• Time Complexity: Consider the efficiency of your solution, especially for larger inputs.

• Input Validation: Ensure the function can handle edge cases, such as zero houses or colors.

Standard Response

Here’s a well-structured sample answer that demonstrates how to write this function:

def num_ways_to_paint_houses(n, k):
 # Edge cases
 if n == 0:
 return 0 # No houses to paint
 if n == 1:
 return k # Only one house can be painted in k ways

 # Initialize the number of ways to paint the first house
 same_color = 0
 diff_color = k

 # Loop through each house from the second to the n-th house
 for i in range(2, n + 1):
 # Calculate ways to paint current house
 new_same_color = diff_color
 new_diff_color = (k - 1) * (same_color + diff_color)

 # Update the same and different color counts
 same_color = new_same_color
 diff_color = new_diff_color

 # Total ways is the sum of same color and different color scenarios
 return same_color + diff_color

# Example usage
houses = 3 # Number of houses
colors = 2 # Number of colors
print(num_ways_to_paint_houses(houses, colors)) # Output: 6

Tips & Variations

Common Mistakes to Avoid

• Ignoring Edge Cases: Failing to account for cases where n or k is zero can lead to incorrect results.

• Overcomplicating the Logic: Keep the logic simple and avoid unnecessary complexity.

• Forgetting to Document: Failing to comment your code can make it difficult for others (or yourself) to understand your solution later.

Alternative Ways to Answer

• Recursive Approach: While dynamic programming is efficient, a recursive approach with memoization can also be used to solve this problem.

• Mathematical Formula: For those familiar with combinatorial mathematics, you may consider deriving a formula to calculate the result directly based on n and k.

Role-Specific Variations

• For a Technical Role: Emphasize the efficiency of your algorithm, discussing time and space complexity in detail.

• For Creative Roles: Highlight the flexibility of the solution and how it can be adapted to various scenarios, perhaps relating it to real-world applications like design choices.

• For Managerial Positions: Discuss the importance of teamwork and collaboration when tackling complex problems like this, and how to ensure everyone is on the same page.

Follow-Up Questions

• How would you modify your function if there were additional constraints, such as a limit on the number of houses painted the same color consecutively?

• Can you provide a scenario where your function might not perform well, and how would you address that?

• What considerations would you take into account if you had to scale your solution for a larger application?

Conclusion

In summary, writing a function to determine the number of ways to paint houses using k colors requires a clear understanding of the problem, effective use of algorithms like dynamic programming, and careful consideration of edge cases. By following the structured approach outlined above, job seekers can better prepare for technical interviews and demonstrate their problem-solving skills effectively