Problem 33 - Algebra 2 and Trigonometry

Question

A manufacturer produces jeans in 9 sizes, 7 different shades of blue, and 6 different leg widths. If a branch store manager orders two pairs of each possible type, how many pairs of jeans will be in stock?

Step-by-Step Solution

Answer

The branch store will have 756 pairs of jeans in stock.

Step 1: Determine the Total Number of Styles

To find out how many different styles of jeans can be produced, we multiply the number of choices for each feature. There are 9 sizes, 7 shades, and 6 leg widths. Therefore, the total number of styles is calculated as follows:\[9 \times 7 \times 6 = 378.\]

Step 2: Account for the Number of Jeans Ordered Per Style

The problem states that the manager orders two pairs of each possible type. Therefore, we need to multiply the total number of styles by 2 to get the total number of pairs:\[378 \times 2 = 756.\]

Key Concepts

Multiplication Principle in Combinatorics

The multiplication principle is a fundamental concept in combinatorics that helps us calculate the total number of ways multiple events can occur. It is a key tool when you're dealing with counting problems that involve several stages or categories, each with a fixed number of options. By multiplying the number of options available in each category, you can determine the total number of possible outcomes.

For example, in our exercise about jeans, the categories are size, shade, and leg width. If you have 9 different sizes, 7 different shades, and 6 different leg widths, you can calculate the number of possible combinations by multiplying these numbers together. This gives us:

• 9 sizes
• 7 shades
• 6 widths

Thus, the total number of styles is calculated as \(9 \times 7 \times 6 = 378\).
This principle simplifies the problem by breaking it into manageable parts, ensuring you systematically account for all possible combinations.

Understanding Permutations

Permutations differ from the multiplication principle, as they account for the specific order of arranging a set of items. With permutations, the order of selection is important, and each arrangement is considered unique.

Suppose you have to arrange different styles of jeans in an order of display; each different order represents a permutation. In problems involving simple choices like our exercise, it's more about combinations than permutations since the order of choosing size, shade, or width doesn't change the style outcome.

So while permutations are a powerful tool elsewhere in combinatorics, the exercise focused primarily on counting combinations using the multiplication principle. Nonetheless, understanding when order matters is crucial for applying permutations accurately.

Solving Counting Problems

Counting problems often involve determining the number of possible ways to select or arrange items. These problems are everywhere, from figuring out how many outfits you can create from a wardrobe to our task of computing how many jeans styles a store can stock.

In our problem, we first determined the number of styles of jeans using the multiplication principle. Then, to solve for the total stock, each style had 2 jeans ordered, leading to:

• Total styles: 378
• Jeans ordered per style: 2

The total number of pairs in stock is calculated as \(378 \times 2 = 756\).
This problem illustrates a basic but essential type of counting problem: one where each option or feature multiplies the total number of outcomes, and additional orders or selections expand the total count further.