Problem 19 - 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(all even-numbered seasons followed by all odd-numbered seasons)

Step-by-Step Solution

Answer

The probability is \( \frac{1}{70} \).

Step 1: Determine Total Arrangements

First, we need to figure out the total number of ways to arrange the 8 DVD cases randomly on the shelf. Since there are 8 cases, the total number of arrangements is the factorial of 8, which is expressed as \[8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320.\]

Step 2: Identify Even and Odd Numbered Seasons

There are 4 even-numbered seasons (2, 4, 6, 8) and 4 odd-numbered seasons (1, 3, 5, 7).

Step 3: Arrange Even-Numbered Seasons Followed by Odd-Numbered Seasons

Consider the scenario where all even-numbered seasons are placed first on the shelf, followed by all odd-numbered seasons. The number of ways to arrange the even-numbered seasons is \[4! = 4 \times 3 \times 2 \times 1 = 24.\]Similarly, the number of ways to arrange the odd-numbered seasons is\[4! = 4 \times 3 \times 2 \times 1 = 24.\]

Step 4: Calculate Probability of Specific Arrangement

The probability of arranging all even-numbered seasons first and then all odd-numbered seasons is the ratio of the number of favorable arrangements to the total number of arrangements, which is \[\text{Probability} = \frac{4! \times 4!}{8!} = \frac{24 \times 24}{40320} = \frac{576}{40320}.\]Simplifying this fraction gives us \[\frac{1}{70}.\]

Key Concepts

Factorial

Factorials are a mathematical concept used to find the total number of ways to arrange or order objects. It is denoted by an exclamation mark (!). For example, when calculating the factorial of a number, you multiply every positive integer up to that number. The factorial of 8, written as \(8!\), is calculated by \[8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320.\] Factorials grow very quickly, which is why they are powerful in combinatorics, where we need to calculate permutations and combinations of objects. Understanding factorials makes it easier to manage these large numbers and facilitates solving complex probability problems. While working with factorials, remember that \(0! = 1\), which often surprises people but is useful in mathematical reasoning.

Permutations

Permutations refer to the different ways in which a set of objects can be arranged or rearranged. The order is important in permutations, which distinguishes them from combinations. In our example with DVD cases, permutations are used to determine how many different ways the seasons can be arranged differently on a shelf. If we want to arrange 4 even-numbered seasons, we calculate it using a factorial, \(4!\), because each of the 4 seasons can be placed in one of the 4 positions, which equals 24 ways. Similarly, the 4 odd-numbered seasons also have \(4!\) permutations, equaling 24 arrangements. Permutations play a crucial role when the sequence of objects affects the outcome, like when determining the order of episodes in a TV series case.

Combinatorics

Combinatorics is the field of mathematics dealing with counting, arrangement, and combination of objects. It helps in solving problems related to probability and is fundamental in determining outcomes in various scenarios.In our DVD case problem, combinatorics helps us figure out the probability of a specific arrangement, namely all even-numbered seasons followed by all odd-numbered seasons. We use the formula for permutations in this context because the order of DVDs is vital. The total arrangement of the DVDs is depicted by the factorial calculation \(8!\), implying all possible ways to sequence 8 items. Combinatorics simplifies complex counting problems and makes it easier to solve probability puzzles by breaking them down into factorials and permutations.

Even and Odd Numbers

Even numbers are integers that are exactly divisible by 2, such as 2, 4, 6, and 8. Odd numbers are integers that are not divisible by 2, leaving a remainder of 1, such as 1, 3, 5, and 7. In problems involving probability, knowing which numbers are even or odd can sometimes influence the arrangement or how items should be grouped, like in our problem of arranging DVD cases. Understanding the difference between even and odd numbers helps when you need to separate or organize data into these two categories for specific probability calculations, such as finding how many ways you can segregate even-numbered seasons from odd-numbered ones. Dealing with even and odd numbers becomes vital when solving probability issues where sequences or arrangements affect the final outcome.