Problem 54 - Algebra 2

Question

REVIEW In a movie theater, 2 boys and 3 girls are seated randomly together. What is the probability that the 2 boys are seated next to each othor? $$ F \frac{1}{5} \quad \text { G } \frac{2}{5} \quad H \frac{1}{2} \quad J \frac{2}{3} $$

Step-by-Step Solution

Answer

The probability is \(\frac{2}{5}\), corresponding to option G.

Step 1: Understand the Total Number of Arrangements

We first need to calculate the total number of ways to arrange 5 people (2 boys and 3 girls) in a line. This can be calculated using the factorial of the total number of people: \(5!\). So, the total number of arrangements is \(5! = 120\).

Step 2: Grouping the Boys Together

To find the probability that the two boys are seated next to each other, we treat them as one "block" or "unit". This means instead of arranging 5 separate people, we arrange 4 units (the block of boys and the 3 girls). The number of ways to arrange 4 units is \(4!\).

Step 3: Arrange the Boys Within the Block

Within this "block" of boys, the two boys can be arranged among themselves in \(2!\) ways. Thus, the number of ways to arrange the boys within the block is 2.

Step 4: Total Favorable Arrangements

Combining steps 2 and 3, the total number of favorable arrangements (where the boys are together) is the product of the number of ways to arrange the 4 units and the number of ways to arrange the boys inside the block: \(4! \times 2! = 24 \times 2 = 48\).

Step 5: Calculate the Probability

The probability that the boys are seated next to each other is the ratio of favorable arrangements to the total arrangements. Therefore, the probability is \(\frac{48}{120} = \frac{2}{5}\).

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and analyzing sets of objects. It's like figuring out how many ways you can mix and match items, such as people sitting in a theater.
When dealing with problems in combinatorics, it's important to understand how many elements are involved and the rules of their arrangement. In our example, we're looking at 2 boys and 3 girls. Combinatorics helps us determine all possible seating arrangements.
Useful techniques in combinatorics include concepts like permutations, combinations, and the use of factorials to find out how items can be arranged or chosen in certain situations.

Factorials

Factorials are a vital concept in combinatorics and probability that describe the product of all positive integers up to a specified number. This is denoted by the symbol "!".
For example, 5 factorial (written as \(5!\)) is equal to \(5 \times 4 \times 3 \times 2 \times 1 = 120\).
This is the total number of ways to arrange 5 people since each person has a unique spot in the line. Factorials are crucial in calculating permutations as they allow us to determine all possible arrangements of a set of items.

Permutations

Permutations refer to the different ways in which a set of items can be ordered. Unlike combinations, permutations focus on arrangements where the order does matter.
In our exercise, we're interested in finding the specific sequence of 5 people where order is crucial because positions matter for who sits next to whom.
For instance, the arrangement of two boys who must sit next together is a permutation problem. By treating them as one unit, we focus on arranging groups in various orders. Using permutations, we can determine how these units shift and rearrange.

Arrangement

Arrangement is a core idea in this type of probability problem. It involves organizing elements in different sequences or patterns according to specific conditions or rules.
In our scenario, we arrange people in a line, mindful of a condition where the 2 boys must sit next to each other.
By treating the boys as a single block and calculating the internal configurations separately, we fine-tune how each group can sit. Considering constraints in arrangement helps us identify all possible setups leading to a particular probability outcome.