A box contains 6 red, 4 blue, and 5 green balls. If three balls are drawn at random without replacement, what is the probability that all three are of different colors? - Coaching Toolbox

Understanding the Context

Why This Ball Set Has Gained Attention Online

The combination of specific color counts (6 red, 4 blue, 5 green) creates a manageable yet insightful probability scenario. Recent social forums and educational content highlight this setup as an accessible gateway to probability theory—especially where rare or limited combinations catch the eye. The controlled depth (only three draws, no repeats) simplifies complex concepts without oversimplifying, making it an effective teachable moment. Moreover, it intersects with trends in STEM education push and digital literacy efforts, where real-world analogies help demystify abstract math. This blend positions the topic at the intersection of curiosity, applied probability, and behavioral psychology—ideal for Discover algorithms favoring relevance, clarity, and depth.

How to Calculate the Probability All Three Draws Are Different Colors

Key Insights

To determine the likelihood that three randomly drawn balls are each of a distinct color, break the problem into manageable parts. First, identify how many ways to pick one red, one blue, and one green ball. With 6 red, 4 blue, and 5 green balls (total 15), the number of favorable combinations is:

6 (red) × 4 (blue) × 5 (green) = 120

But since the order of drawing doesn’t matter, we count combinations, not permutations: (6C1 × 4C1 × 5C1) = 6 × 4 × 5 = 120 distinct color-matched draws.

Next, calculate total possible three-ball draws from 15: using the combination formula C(n, k) = n! / (k!(n−k)!), the total combinations are:

C(15, 3) = 455

Final Thoughts

So, the probability is 120 ÷ 455, which simplifies to approximately 0.2637—about 26.37%. This meaningful ratio illustrates how simple setups can yield rich educational opportunities, fitting modern users’ desire to engage thinking deeply while swiping through intuitive, fast-loading content.

Common Questions Readers Are Asking

People drawn to this problem often hope to understand how random chance works in tangible scenarios. Here’s how to clarify their intent:

• How do you count combinations with constraints?
  Confirming order matters only in wording, not outcome—we group outcomes and divide by total unordered sets, building foundational combinatorics intuition.

• Does drawing without replacement change results?
  Yes—after the first ball is drawn, fewer remain, reducing future probabilities. This dependency models real-world systems like gaming odds or sampling methods.