Problem 17 - Algebra and Trigonometry

Question

A girl has 5 skirts, 8 blouses, and 12 pairs of shoes. How many different skirt-blouse-shoe outfits can she wear? (Assume that each item matches all the others, so she is willing to wear any combination.)

Step-by-Step Solution

Answer

There are 480 different skirt-blouse-shoe outfits possible.

Step 1: Understand the Problem

We need to determine the total number of unique outfits consisting of one skirt, one blouse, and one pair of shoes. Each skirt can be paired with any blouse and any pair of shoes.

Step 2: Calculate Combinations for Blouses and Skirts

For each skirt, there are 8 different blouses to choose from. Since there are 5 skirts, the total number of skirt-blouse combinations is calculated by multiplying 5 by 8.\[5 \times 8 = 40\]

Step 3: Calculate Total Outfits with Shoes

Each skirt-blouse combination can be matched with any of the 12 pairs of shoes. Therefore, multiply the number of skirt-blouse combinations by the number of shoe pairs to find the total number of different outfits.\[40 \times 12 = 480\]

Step 4: Conclude Total Outfit Combinations

The total number of different skirt-blouse-shoe outfits is 480, which includes every possible combination of skirts, blouses, and shoes.

Key Concepts

multiplication principle

The multiplication principle, also known as the fundamental principle of counting, is a key concept in combinatorics. It allows us to find the total number of combinations or outcomes when dealing with multiple choices.

Consider it as a method to calculate how many ways you can combine different items or categories. In essence, if you have multiple categories and each category has a certain number of options, the multiplication principle states that you can determine the total number of outcomes by multiplying the number of options available in each category.

For example, if you're choosing an outfit, you might have several choices for shirts, pants, and shoes. By applying the multiplication principle, you can determine the total number of possible outfits simply by multiplying the number of shirts by the number of pants and then by the number of shoes.

This approach is not limited to clothing - it applies to any scenario where you select one item from each of several categories.

outfit combinations

Creating outfit combinations can be a fun and practical application of the multiplication principle. It involves selecting one item from each category to create a complete set or outfit.

In our example with a girl who has several skirts, blouses, and pairs of shoes, we're determining how many distinct outfits she can create. It's as simple as choosing one skirt, one blouse, and one pair of shoes. Each choice is independent of the others, which is why every item goes with every other item.

• First, choose the skirt. You have 5 options.
• Next, select the blouse. There are 8 options.
• Finally, pick a pair of shoes, with 12 options available.

Thus, the total number of combinations is 5 times 8 times 12, resulting in 480 unique outfits.

This kind of calculation is typically straightforward but serves as an essential skill for understanding larger and more complex combinatorial problems.

problem solving

Problem solving in combinatorics often involves breaking down a large problem into manageable steps. Understanding the task and identifying which principles apply is crucial.

In the given problem, we know we need to figure out the possible combinations of skirts, blouses, and shoes. By clearly defining each component of the outfit separately, we can then apply the appropriate mathematical principle to find the solution.

The steps include:

• Understand the problem: Identify the variables. In this case, skirts, blouses, and shoes.
• Calculate combinations for two categories: Multiply options for skirts by blouses.
• Integrate the third category: Multiply the result by the number of shoe options.

This structured approach simplifies complex problems and makes them easier to solve. Always double-check the final number to ensure it accounts for all possible combinations based on the given parameters.