Brooch Selection: Math Problem For 14 Women

Hey guys, let's dive into a fun math problem that involves a group of 14 women from an RT (neighborhood association) preparing for an event at the Kecamatan Office. They've got these awesome red brooches to wear, and there are 4 different types available. The challenge? Figuring out how many ways they can choose these brooches. This is a classic combinatorics problem, and we're going to break it down step by step. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into calculations, let’s make sure we fully understand the scenario. We have a group of 14 women, each of whom needs to wear a brooch. There are 4 different types of red brooches to choose from. The question we need to answer is: How many different ways can these 14 women select brooches from the 4 available types? This is a combinatorial problem, which means we're looking at combinations and arrangements rather than simple arithmetic. Combinatorics is all about counting possibilities, and it pops up in many real-world scenarios, from scheduling events to designing experiments. It’s a branch of math that helps us understand how many different outcomes are possible when we have a set of choices. In our case, the choices involve selecting brooches, and we want to know the total number of ways these choices can be made by the 14 women. This problem assumes that each woman can choose any of the 4 types of brooches, and different women can choose the same type. We're essentially distributing the brooches among the women, considering the variety of types available. The complexity arises from the fact that we're not just picking one item; each woman is making a choice, and we need to account for all possible combinations of these choices. Understanding the problem clearly is the first and most crucial step in finding the solution. Once we know exactly what we're trying to calculate, we can start looking at the appropriate formulas and methods to get there. So, let's keep this scenario in mind as we move forward and explore the mathematical tools that will help us solve it.

Identifying the Right Approach

To solve this problem, we need to identify the right mathematical approach. Since each of the 14 women can choose from 4 different types of brooches independently, this is a problem of combinations with repetition. In other words, each woman's choice doesn't affect the others, and they can all choose the same type of brooch if they want. This type of problem falls under the umbrella of combinatorics, specifically dealing with combinations when repetition is allowed. Combinations with repetition are different from regular combinations where you can't pick the same item more than once. Think of it like this: if you're picking lottery numbers, you can't pick the same number twice, but in our brooch scenario, multiple women can certainly choose the same type of brooch. The formula for combinations with repetition is given by:

C(n + r - 1, r)

Where:

• n is the number of types of items (in our case, the number of types of brooches, which is 4).
• r is the number of items to choose (in our case, the number of women, which is 14).
• C(x, y) represents the number of combinations of x items taken y at a time, also known as "x choose y".

This formula helps us account for all the different ways the women can choose the brooches, considering that they can repeat their choices. It’s a powerful tool in combinatorics that allows us to solve problems where the order of selection doesn't matter, but the repetition does. Using this formula, we can plug in the values for our problem and calculate the number of different ways the women can choose their brooches. It's important to use the correct formula, as using a standard combination formula (without repetition) would give us the wrong answer. So, with the formula in mind, let's move on to the calculation step and see how we can apply it to find the solution.

Performing the Calculation

Now that we've identified the correct formula, let's plug in the numbers and do the calculation. We have:

• n = 4 (the number of types of brooches)
• r = 14 (the number of women)

Using the formula for combinations with repetition:

C(n + r - 1, r) = C(4 + 14 - 1, 14) = C(17, 14)

C(17, 14) means "17 choose 14", which is the number of ways to choose 14 items from a set of 17. The formula to calculate C(x, y) is:

C(x, y) = x! / (y! * (x - y)!)

Where ! denotes the factorial function (e.g., 5! = 5 × 4 × 3 × 2 × 1). So, for our problem:

C(17, 14) = 17! / (14! * (17 - 14)!) = 17! / (14! * 3!)

Let's break this down. 17! is 17 × 16 × 15 × ... × 1, 14! is 14 × 13 × ... × 1, and 3! is 3 × 2 × 1 = 6. We can simplify the expression by canceling out the common terms in the numerator and the denominator:

C(17, 14) = (17 × 16 × 15 × 14!)/(14! × 3 × 2 × 1) = (17 × 16 × 15) / (3 × 2 × 1)

Now, we can further simplify:

C(17, 14) = (17 × 16 × 15) / 6 = 17 × 8 × 5 = 17 × 40 = 680

So, C(17, 14) = 680. This means there are 680 different ways for the 14 women to choose the red brooches from the 4 available types. The calculation involves understanding factorials and how to simplify the combination formula to arrive at a numerical answer. It’s a blend of arithmetic and combinatorial principles, and it shows how we can use mathematical tools to solve real-world problems involving choices and arrangements. Now that we have our final answer, let's summarize our findings and understand the significance of this result in the context of the problem.

Final Answer

After performing the calculation, we found that there are 680 different ways for the 14 women to choose the red brooches from the 4 available types. This result tells us the sheer number of possible combinations when each woman can independently select a brooch type, with repetition allowed. In practical terms, this means that the women have a wide variety of options when choosing their brooches, leading to a diverse set of possible outcomes. The number 680 represents the total number of unique distributions of the 4 types of brooches among the 14 women. Each of these 680 combinations represents a different scenario in which the women could choose their brooches, considering the variety of types available. This is a significant number, highlighting the complexity that can arise from seemingly simple choices when multiple individuals are involved. Understanding the final answer helps us appreciate the scope of combinatorial possibilities and how they can apply to real-world situations. Whether it's planning events, distributing resources, or making selections from a set of options, combinatorics provides the tools to analyze and quantify these possibilities. So, in our brooch selection problem, the 680 different ways represent the rich tapestry of choices available to the women, and it showcases the power of combinatorics in action. Hope you guys enjoyed this math adventure!