Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

Wardrobe mix ' \(n\) ' match A girl has four skirts and six blouses. How many different skirt-blouse combinations can she wear?

Step-by-Step Solution

Answer

There are 24 different skirt-blouse combinations.

Step 1: Understand the Problem

We need to find out how many different combinations of skirts and blouses the girl can wear. Available are 4 skirts and 6 blouses.

Step 2: Calculate Possible Combinations

Each skirt can be paired with any of the 6 blouses, allowing for multiple combinations. Thus, we multiply the number of skirts by the number of blouses to find the total combinations.

Step 3: Use the Formula for Combinations

The number of combinations is calculated by multiplying the number of skirts by the number of blouses: 4 skirts * 6 blouses = 24 combinations.

Key Concepts

multiplication principle

The multiplication principle is a cornerstone concept in combinatorics, which helps simplify complex counting problems. It states that if you have two independent choices, where one choice can be made in "n" ways and the other in "m" ways, the total number of ways to make both choices is the product of "n" and "m".
For instance, consider the problem of pairing skirts and blouses. A girl has 4 different skirts and 6 different blouses. According to the multiplication principle, the total number of possible skirt-blouse combinations can be found by multiplying the number of skirts (4) by the number of blouses (6).
Thus, the answer is 24 combinations.

• This principle is widely used to solve problems where multiple independent choices need to be counted.
• The idea is based on calculating all possible outcomes by considering every outcome from each set.

It is fundamental in defining and solving numerous counting challenges not just in daily life but also in more complex fields like computer science and probability.

permutations

Permutations relate to arrangements within combinatorics. Specifically, they focus on the order of arranging a set of items. In permutations, the sequence in which the items are arranged matters.
For example, consider how you can arrange the letters A, B, and C. If you want to know how many ways you can order these three letters, you are dealing with permutations.

• The number of permutations of a set of "n" items is given by "n!", which means "n factorial," calculated as the product of all positive integers up to "n".
• For 3 items, it is calculated as 3! = 3 × 2 × 1 = 6 possible arrangements.

It is important to remember that, unlike combinations, permutations are concerned with the specific arrangement or sequence of elements, making them crucial when the order of items is significant.

counting problems

Counting problems are essential components of combinatorics, involving determining the number of ways patterns or sets can be arranged or selected. They serve as the basis for solving problems involving arrangements, selections, and combinations of items or objects.
Take, for example, the wardrobe problem given earlier. Here, the challenge is to count the number of possible outfit combinations using multiplication. This illustrates a straightforward counting problem.

• Counting problems can vary from simple to complex, but often rely on basic principles like the addition and multiplication principles.
• Understanding whether an order matters (permutations) or doesn't matter (combinations) is essential to solving these problems.

These problems often form the basis of probability calculations, addressing real-world scenarios where determining exact outcomes is crucial.