Question

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

The minimum number of colors needed is 6.

Step-by-step solution

- Understand the problem

We need to determine the minimum number of colors required to code 20 chemicals uniquely with either single colors or pairs of colors, where the order of the colors in the pairs does not matter.

- Define combinations

If there are n colors, each color can be used standalone (single-color combinations), and any two different colors can be used in pairs. The total number of unique combinations is therefore the sum of the single-color combinations and the two-color combinations.

- Calculate the formula

The number of single-color combinations is n. The number of unique two-color combinations is given by the binomial coefficient \(\binom{n}{2}\), which equals \(\frac{n(n-1)}{2}\). Thus, the total number of combinations is \(n + \binom{n}{2}\).

- Set up the equation

We need this total number of combinations to be at least 20. Therefore, we need to solve the inequality \(n + \frac{n(n-1)}{2} \geq 20\).

- Simplify and solve the inequality

Rewriting the inequality gives \(n + \frac{n^2 - n}{2} \geq 20\). Multiplying every term by 2 to clear the fraction, we get \(2n + n^2 - n \geq 40\), which simplifies to \(n^2 + n - 40 \geq 0\).

- Solve the quadratic inequality

Using the quadratic formula \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) for \(n^2 + n - 40 = 0\), we get \(n = \frac{-1 \pm \sqrt{1 + 160}}{2} = \frac{-1 \pm 13}{2}\). This gives us n = 6 or n = -7. Since the number of colors cannot be negative, we have n = 6.

- Verify the solution

With 6 colors: single-color combinations = 6, two-color combinations = \(\binom{6}{2} = 15\). Therefore, the total number of combinations is 6 + 15 = 21, which is sufficient.

Key concepts

Color-Coding System

In a laboratory setting, chemicals can be identified using various methods. One practical approach is through a color-coding system. This system assigns specific colors or combinations of colors to represent different chemicals. By organizing and simplifying chemical identification, scientists can easily differentiate between substances at a glance.
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

To determine the total number of unique combinations of colors, we can use the binomial coefficient which is essential in combinatorial mathematics. The binomial coefficient, denoted as \(\binom{n}{k}\), represents the number of ways to choose k elements from a set of n elements without regard to the order.
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

A quadratic equation is an essential concept in algebra and is used in this exercise to determine the minimum number of colors needed for the coding system. It is an equation of the form \(ax^2 + bx + c = 0\).
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

Chemical identification in laboratories is crucial for safety and accuracy in experiments. Using a systematic approach, such as a unique color-coding system, enhances efficiency and reduces the potential for error.
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

Pair combinations come into play when we want to use two colors to represent a chemical, where the order of colors does not matter. An essential concept here is that each pair should be counted only once, thus considering them as combinations rather than permutations.
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.