Problem 22 - College Algebra

Question

Use the Fundamental Counting Principle to solve Exercises \(21-32\). A popular brand of pen is available in three colors (red, green, or blue) and four writing tips (bold, medium, fine, or micro). How many different choices of pens do you have with this brand?

Step-by-Step Solution

Answer

Total number of different pen choices = 12

Step 1: Identify the number of different pen colors

From the exercise, the pen is available in three different colors which are red, green and blue.

Step 2: Identify the different types of writing tips

The exercise also states that the pen is available in four writing tips which are bold, medium, fine, and micro.

Step 3: Apply the fundamental counting principle

According to the Fundamental Counting Principle, the total number of different choices of pens with this brand can be obtained by multiplying the number of different colors of pens and the number of different writing tips. \n Therefore, the total number of different pens = \( 3 \text{ colors} \) * \(4 \text{ writing tips} \)

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that focuses on counting, arranging, and grouping objects. It helps us determine how to combine different sets of elements. In the context of our pen selection problem, combinatorics allows us to figure out how many different combinations of pen colors and tips we can have. By learning about combinatorics, you can easily solve problems that involve making choices or creating arrangements.
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When you're trying to count combinations, you often consider aspects like:

• The number of ways to choose items from a set.
• Whether the order of choices matters.
• Whether you can repeat items in your choices.

For the pen selections, we think about each choice separately (colors and tips) and then decide how to combine them using the simpler principles of counting, such as multiplication, which we'll explore next.

Multiplication Principle

The multiplication principle, also known as the fundamental counting principle, is a vital part of combinatorics. It states that if you have several independent choices to make, you can find the total number of combinations by multiplying the number of options for each choice. This principle works because each choice is made independently of the others.

In terms of our pen example, the multiplication principle tells us:

• We have 3 choices for the pen color: red, green, or blue.
• We have 4 choices for the writing tip: bold, medium, fine, or micro.

To find the total set of combinations, you multiply these two numbers:
The total number of different pen choices = \( 3 \text{ colors} \times 4 \text{ tips} = 12 \).

This method is systematic and straightforward, reducing complex problems into simple arithmetic.

Permutations

Permutations refer to the different ways in which a set of objects can be ordered or arranged. However, it’s important to note that in permutations, the order of arrangement matters, unlike in our pen problem where only the combination of features counts.

In situations where order is important, you calculate permutations. For example, rearranging the colors and tips would involve checking every possible arrangement for a different outcome.

Permutations are crucial when arranging items or tasks in a specific sequence or when specific orders make a difference. Although not directly used in the pen count in this problem, understanding permutations gives a broader view of combinatorial problems, helping to decide when you need to count arrangements versus combinations.