Problem 18 - Algebra 2

Question

Janice has 8 DVD cases on a shelf, one for each season of her favorite TV show. Her brother accidentally knocks them off the shelf onto the floor. When her brother puts them back on the shelf, he does not pay attention to the season numbers and puts the cases back on the shelf randomly. Find each probability. P(seasons 1 through 4 in the correct positions)

Step-by-Step Solution

Answer

The probability is \( \frac{1}{1,680} \).

Step 1: Understand the Total Number of Arrangements

There are 8 DVD cases, and each can be placed into any slot on the shelf. Thus, there are a total of 8! ("8 factorial") ways to arrange these cases. The calculation for 8! is:\[8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40,320\] So, there are 40,320 total arrangements of the DVD cases.

Step 2: Identify Desired Arrangements

We need the specific cases for seasons 1 through 4 to be in the correct order. Once these four are correctly placed, the remaining four seasons (5 to 8) can be in any order. Therefore, the order of seasons 5 to 8 is the only variable.This means we fix seasons 1-4 in their correct positions and arrange seasons 5-8. There are 4! (4 factorial) ways to arrange the remaining four cases:\[4! = 4 \times 3 \times 2 \times 1 = 24\] So, there are 24 ways to arrange the remaining cases once seasons 1-4 are correctly placed.

Step 3: Calculate the Probability

The probability that seasons 1-4 are all correctly positioned is the ratio of the number of favorable arrangements to the total arrangements.The probability formula is:\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{24}{40,320} = \frac{1}{1,680} \]Therefore, the probability is \( \frac{1}{1,680} \).

Key Concepts

Factorial

In mathematics, a factorial is a function that multiplies a series of descending natural numbers. It is denoted by an exclamation mark "!" after the number. For example, the factorial of the number 8, represented as 8!, means you multiply 8 by every whole number smaller than itself down to 1. So, 8! is calculated as follows:

\[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40,320 \]

This concept helps us understand the total number of ways to arrange a set of items, like the DVD cases on Janice's shelf. Each unique sequence of arranging these items is called a "permutation." The factorial tells you how many such permutations exist.

Arrangements

Arrangements refer to the different orders in which a set of items can be organized. In our example with the DVD cases, an arrangement would indicate a possible sequencing of seasons on the shelf. Think of arrangements like a playlist order: each order is distinct even if the same items are used.

For Janice's 8 DVDs, using factorials, we calculated that there were 40,320 different arrangements. If she only cared about placing DVDs 1 through 4 correctly, then the different ways to arrange DVDs 5 through 8 mattered. That was computed as 4!, resulting in 24 different arrangements.

Combinatorics

Combinatorics is a branch of mathematics essential for understanding probabilities in cases like Janice's. It deals with figuring out how many ways you can choose and arrange items from a set. It's very helpful for solving probability problems by counting how many arrangements are possible and how many meet the desired criteria.

When calculating the probability of an event, combinatorics allow us to identify both the total possible arrangements and the "favorable" ones—arrangements that meet specific conditions, like the first four DVD seasons being in the correct order. This is how we arrived at calculating a probability of 1 in 1,680 for our earlier exercise.

Favorable Outcomes

Favorable outcomes are specific instances where the conditions of a problem are met. In probability exercises, these are the successful events that satisfy what you are solving for.

For Janice's DVD arrangement, the favorable outcomes were those sequences where DVDs for seasons 1 through 4 were in the correct order. This problem considered how many ways we could arrange seasons 5 through 8 once seasons 1 through 4 were fixed. The number of these arrangements is found using factorials. Since there are 24 such arrangements, the probability then equals the ratio of favorable outcomes to the total outcomes.

• Favorable Outcomes: 24
• Total Arrangements: 40,320

Calculating this ratio gave us a probability of 1/1,680.