Problem 34 - Algebra 2 and Trigonometry

Question

In \(32-37,\) determine the number of different arrangements. Stacking 6 red, 4 yellow, and 2 blue t-shirts

Step-by-Step Solution

Answer

There are 34,650 different arrangements of the t-shirts.

Step 1: Understand the Problem

We need to find how many different ways we can arrange a stack of 6 red, 4 yellow, and 2 blue t-shirts. These t-shirts must be considered as distinguishable items only by color, not by individual identity.

Step 2: Calculate Total Arrangements

First, calculate the total number of arrangements of 12 t-shirts if they were all distinguishable. The formula for arranging \( n \) distinct items is \( n! \). Here, \( n = 6 + 4 + 2 = 12 \), so we calculate \( 12! \).

Step 3: Adjust for Indistinguishable Items

Since the t-shirts of the same color are indistinguishable, adjust the total arrangements by dividing by the factorial of the quantity of each color: \( \frac{12!}{6! \times 4! \times 2!} \). This accounts for repeated items.

Step 4: Perform the Calculations

Calculate each factorial: \( 12! = 479001600 \), \( 6! = 720 \), \( 4! = 24 \), and \( 2! = 2 \).Now compute the number of different arrangements:\[\frac{479001600}{720 \times 24 \times 2} = 34650\]

Step 5: Conclude the Problem

There are 34,650 different ways to arrange the stack of t-shirts.

Key Concepts

Factorial

In combinatorics, a factorial is a mathematical operation that is essential for calculating permutations or arrangements of a set of items. The factorial of a number, denoted by an exclamation mark (e.g., \( n! \)), is the product of all positive integers less than or equal to that number. So, \( n! = n \times (n-1) \times (n-2) \times \, \ldots \, \times 1 \). For instance, the factorial of 4 is calculated as follows:

• \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)

Factorials are used to compute permutations of a set of distinct items. If you want to know how many ways you can arrange 12 different t-shirts in a stack, you would use \( 12! \) to get the total number of distinct permutations. In the case of the t-shirt problem, we first calculated \( 12! \) to find all the possible arrangements if each t-shirt were unique. This value, however, will later be adjusted to account for the indistinguishable nature of the t-shirts of the same color.

Indistinguishable Objects

The concept of indistinguishable objects in combinatorics deals with situations where some items are not distinct from each other. For example, in the problem of arranging t-shirts, we have 6 red, 4 yellow, and 2 blue t-shirts that are identical within their respective colors. When counting arrangements, this lack of distinction among similar items means that arrangements such as changing two red t-shirts will not create a new unique arrangement. This affects the total count of arrangements because those switches don't lead to new outcomes.To adjust for the indistinguishability of items, the factorial of the number of indistinguishable objects is used to divide the total number of arrangements. For the t-shirt problem:

• The total number of arrangements without regard to color is \( 12! \).
• Divide by \( 6! \) for the red t-shirts.
• Divide by \( 4! \) for the yellow t-shirts.
• Divide by \( 2! \) for the blue t-shirts.

This helps to find the true number of unique arrangements, which are not distinct within their color groups.

Arrangement Problems

Arrangement problems in combinatorics are focused on determining the number of ways a set of items can be arranged. This is a fundamental problem in counting and requires understanding of permutations and combinations. When dealing with arrangement problems, especially those that involve indistinguishable items, you need to carefully consider how the identity of each object—whether or not they are distinct—affects the count. The formula used in such situations is:\[ \frac{n!}{n_1! \times n_2! \times \ldots \times n_k!} \]where \( n \) is the total number of items, and \( n_1, n_2, \ldots, n_k \) represent the groups of indistinguishable items. In our example, despite having 12 t-shirts, because they consist of indistinguishable collections of colors, the permutation calculation reduced to dividing by each color's factorial. This allows us to find a more accurate count of possible arrangements without overcounting identical configurations.Understanding how to adjust for indistinguishable items is crucial in tackling more advanced arrangement problems. This knowledge enables you to solve complex problems involving non-unique items effectively.