Question: An industrial designer offers custom phone cases with 5 color palettes and 4 pattern options. A customer picks exactly 3 different colors and 1 pattern, where the order of colors does not matter. How many unique case designs can be made? - Parker Core Knowledge

Understanding the Context

Why Custom Case Design is Resonating Across the U.S.

In a digital age where personalization is key, consumers increasingly expect products tailored to their tastes. This demand mirrors broader cultural shifts—mobile users seek distinct, meaningful ways to express identity through smartphone accessories. With growing interest in niche aesthetics, limited-edition design options have become a powerful differentiator for brands. From vibrant, earthy tones to sleek gradients, users want control over every detail, turning phone cases into subtle statements of style. The ability to select exactly three colors from five — without fixating on placement — ensures both ease and creativity, making this experience increasingly popular among discerning buyers.

How the Design Logic Works: A Clear Breakdown

Key Insights

To determine the total number of unique custom phone case designs, we break the problem into two key choices: color and pattern.

First, selecting 3 colors from a palette of 5. Since order doesn’t matter, this is a combination problem. The formula C(5,3) calculates how many ways we choose 3 items without regard to sequence:
C(5,3) = 5! / (3! × (5–3)!) = 10
So, there are 10 unique ways to choose the colors.

Second, choosing a pattern. There are 4 distinct pattern options, and only one pattern can be selected per case. Each color combination is paired with just one pattern, offering 4 variations independent of color choices.

Multiplying both components gives the total number of distinct designs:
10 color combinations × 4 pattern options = 40 unique case designs

Final Thoughts

Common Questions About Custom Design Choices

H3: How are color combinations calculated?