Chapter 3: Problem 37
In a certain laboratory, chemicals are identified by a color-coding system. There are 20 different chemicals. Each is coded with either a single color or a unique two-color combination. If the order of colors in the pairs doesn't matter, what is the minimum number of different colors needed to code all 20 chemicals with either a single color or a unique pair of colors?
Short Answer
Step by step solution
- Understand the problem
- Define combinations
- Calculate the formula
- Set up the equation
- Simplify and solve the inequality
- Solve the quadratic inequality
- Verify the solution
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Color-Coding System
In this exercise, we have 20 chemicals, and we aim to identify each uniquely using either a single color or a unique combination of two colors where the order doesn't matter. This ensures we avoid confusion and maintain clarity in laboratory processes.
Binomial Coefficient
In the exercise, if there are n colors available, then each color can be used alone (single-color) or in pairs (two-color combinations) without considering the order. The number of two-color combinations is given by \(\binom{n}{2} = \frac{n(n-1)}{2}\).
Quadratic Equations
Here, our task converts into solving the inequality \(n + \frac{n(n-1)}{2} \geq 20\), which simplifies to a quadratic equation \(n^2 + n - 40 \geq 0\). Solving this using the quadratic formula \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), we find n = 6 as the smallest positive integer that meets the requirements.
Chemical Identification
In real-world applications, a well-thought-out identification system can prevent accidents and ensure that chemicals are handled correctly. For the 20 chemicals in question, coding with either a single color or a pair of colors aids in quick recognition and simplifies the management of chemical substances.
Pair Combinations
For example, if we have 6 colors, the number of unique pairs is calculated using the formula \(\binom{6}{2} = 15\). Adding the single-color combinations (6 in this case), we find a total of 21 unique ways to code the chemicals, ensuring that all 20 can be distinctly identified.