Problem 16 - 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(season 5 in the correct position)

Step-by-Step Solution

Answer

The probability is 1/8 that season 5 is in the correct position.

Step 1: Understanding the Problem

We need to calculate the probability that the DVD case for season 5 is placed in the correct position on the shelf. There are a total of 8 DVD cases, each for a different season, and the cases are placed randomly back on the shelf.

Step 2: Total Number of Arrangements

Since there are 8 DVD cases, the total number of ways to arrange these cases is the factorial of 8. Therefore, we calculate this as: \[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40,320 \]This gives us the total number of possible arrangements of all 8 cases.

Step 3: Number of Favorable Outcomes

For the probability that the season 5 case is in the correct position, we only care about that single case being correctly placed. The remaining 7 cases can be in any order. The number of ways to arrange the remaining 7 cases is: \[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5,040 \]Thus, there are 5,040 favorable outcomes where the season 5 case is in the correct position.

Step 4: Calculating the Probability

The probability that the season 5 case is correctly placed is the ratio of favorable outcomes to the total number of outcomes. Therefore, the probability is:\[ P(\text{season 5 in correct position}) = \frac{7!}{8!} = \frac{5,040}{40,320} = \frac{1}{8} \]

Step 5: Conclusion

The calculated probability shows that there is a 1 in 8 chance that the season 5 DVD case randomly falls back into its original position.

Key Concepts

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, arranging, and analyzing configurations of objects. It is often used in probability theory to determine the likelihood of various combinations or arrangements. In scenarios where you have a set number of objects and you want to find out how they can be arranged or selected, combinatorics provides the tools to calculate these possibilities.

• Imagine you're trying to find the number of ways to arrange a set of objects. Combinatorics tells you how to count these arrangements efficiently.
• When determining probabilities, like Janice's scenario, we often list all possible combinations or arrangements to identify favorable outcomes.

In the context of the problem where Janice's DVDs are jumbled and placed randomly, combinatorics helps us to calculate both the total number of arrangements and the specific scenarios where a DVD might end up in its original spot.

Factorial

A factorial, denoted by an exclamation mark (e.g., 8!), represents the product of an integer and all the positive integers below it. Factorials are fundamental in calculations involving permutations and combinations. They allow us to easily compute the total number of ways to arrange a set of objects.

• The factorial of a number \( n \) is the product of all positive integers less than or equal to \( n \).
• For example, \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \).
• In Janice's problem, \( 8! \) represents the total number of ways to arrange 8 distinct DVD cases on a shelf.

By using factorials, we quickly understand how extensive the possibilities are when order matters. They provide a structured way to count arrangements without having to list each one manually.

Permutations

Permutations refer to arrangements of objects in a specific order. When order matters, permutations are what we use to count the possible configurations. A permutation focuses not just on which items are selected but on the sequence they occur in.

• In Janice's case, if her brother randomly places the DVDs back on the shelf, each unique ordering of those DVDs is a permutation.
• For 8 DVD cases, the number of permutations is calculated as \( 8! \), representing all possible sequences.
• "Favorable outcomes" in Janice’s situation includes any sequence where the season 5 DVD lands correctly, accommodating all permutations of the other 7.

Understanding permutations helps solve problems where the arrangement sequence directly impacts the outcome, as it does in Janice’s probability question.