Problem 4 - Algebra 2

Question

An administrative assistant has 4 blue file folders, 3 red folders, and 3 yellow folders on her desk. Each folder contains different information, so two folders of the same color should be viewed as being different. She puts the file folders randomly in a box to be taken to a meeting. Find each probability. P(4 blue, 3 red, 3 yellow, in that order)

Step-by-Step Solution

Answer

The probability is \( \frac{1}{5040} \).

Step 1: Determine Total Number of Folder Arrangements

We begin by calculating the total number of arrangements in which the folders can be placed in the box. There are 4 blue folders, 3 red folders, and 3 yellow folders. The total number of folders is 4 + 3 + 3 = 10 folders. The total arrangements are given by the factorial of the total number of folders: \[ 10! \]

Step 2: Calculate Arrangements of Folders by Color

Next, we need to account for the arrangements of folders within each color since they are distinguishable only by order. The number of arrangements for the blue folders is \[ 4! \]; for the red folders is \[ 3! \]; and for the yellow folders is \[ 3! \]. This accounts for the internal ordering of folders of each color.

Step 3: Use the Multinomial Formula to Find Probabilities

To find the probability of placing the folders in the specified order (4 blue, 3 red, 3 yellow), we use the multinomial formula, which is given by: \[ \frac{10!}{4! \times 3! \times 3!} \]. This calculation gives us the number of favorable outcomes (which is 1, as the order is fixed) divided by the total possible outcomes (the above result).

Step 4: Calculate the Probability of the Specific Arrangement

Since there is only 1 way to specifically order the folders as 4 blue followed by 3 red and then 3 yellow, the probability is \[ \frac{1}{\frac{10!}{4! \times 3! \times 3!}} \]. Calculate the factorials to obtain the probability.

Key Concepts

Multinomial Formula

The multinomial formula is a crucial concept in probability and combinatorics used to count the number of ways to distribute objects into categories of varying sizes. It generalizes the idea of permutations and combinations for multisets, where objects can be grouped and counted differently based on their categories.
To understand this formula, consider its difference from simpler cases of arranging distinct objects. The multinomial formula calculates the number of ways to arrange these objects when you have multiple categories, each containing indistinguishable items.
When applied, the formula looks like this:

• N is the total number of objects, drawn from different categories.
• n_1, n_2, ..., n_k represents the number of objects in each category.

The multinomial formula is:\[\frac{N!}{n_1! \times n_2! \times ... \times n_k!}\]In our example with file folders, it helps determine how many possible ways can be arranged based on their color, giving the ratio of desired arrangements relative to all possible arrangements.

Factorial

Factorials are used to calculate the total number of ways to arrange objects. Denoted by an exclamation point (!), a factorial of a number is the product of all positive integers up to that number. For example, the factorial of 5, written as \(5!\), is calculated as:

• 5 × 4 × 3 × 2 × 1 = 120

Factorials grow rapidly as the number increases, which makes them a powerful tool in combinatorics and permutations.
In this exercise, calculating \(10!\) enables us to determine how many total arrangements are possible for the folders. Each category (color) of folders also needs its own factorial calculation: \(4!\) for blue, \(3!\) for red, and \(3!\) for yellow.
These individual factorial calculations work together in the multinomial formula to account for varying numbers of indistinguishable objects in each category, helping to determine the probability of a specific order.

Combinatorics

Combinatorics refers to the branch of mathematics concerning the counting, arrangement, and combination of objects.
It helps solve problems related to permutations and combinations, which are integral to understanding probability.
In this scenario, when trying to determine the probability of a specific order of file folders, combinatorics provides the tools needed to navigate through different possibilities. By evaluating all possible arrangements and narrowing them down to the genuine cases that match our condition (4 blue, 3 red, 3 yellow), we can discern the situation's likelihood.
Using combinatorics, we look at every possible case and filter out those that meet specific criteria, in this instance by employing the multinomial theorem. The approach involves:

• Calculating the range of all potential arrangements
• Determining how distinct grouping impacts these arrangements
• Finally narrowing it down to favorable cases

Thus, using these foundational tools, we can calculate probabilities and understand complex distribution and arrangement patterns.