Understanding the Problem: Probability in Arrangements

The question asks for the probability that three boys sit together when six people (three boys and three girls) are arranged in a row for a group photograph. To find the probability, we need two values:

1. The total number of possible arrangements of the six people.
2. The number of arrangements where all three boys sit together (favorable arrangements).

Probability is then calculated as:

\(\text{Probability} = \frac{\text{Number of favorable arrangements}}{\text{Total number of possible arrangements}}\)

Calculating Total Possible Arrangements

We have a total of 6 distinct people (3 boys and 3 girls) to arrange in a row. The number of ways to arrange \(n\) distinct items in a row is given by \(n!\) (n factorial).

In this case, \(n=6\).

Total number of arrangements \(= 6!\)

\(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\)

So, there are 720 total possible ways to arrange the six people in a row.

Calculating Favorable Arrangements: Boys Sitting Together

We want to find the number of arrangements where all three boys (P, Q, R) sit together. To handle this condition, we can treat the group of three boys as a single unit.

Imagine tying the three boys together. Now, instead of 6 individual people, we are arranging 4 entities:

• The unit of 3 boys
• Girl S
• Girl T
• Girl U

The number of ways to arrange these 4 entities in a row is \(4!\).

\(4! = 4 \times 3 \times 2 \times 1 = 24\)

However, within the unit of three boys, the boys themselves can be arranged in different orders. The three boys (P, Q, R) can arrange themselves within their unit in \(3!\) ways.

\(3! = 3 \times 2 \times 1 = 6\)

To find the total number of arrangements where the three boys sit together, we multiply the number of ways to arrange the 4 entities by the number of ways the boys can arrange themselves within their unit.

Number of favorable arrangements = (Arrangement of the 4 entities) \(\times\) (Arrangement of boys within the unit)

Number of favorable arrangements \(= 4! \times 3! = 24 \times 6 = 144\)

So, there are 144 arrangements where all three boys sit together.

Calculating the Probability

Now we can calculate the probability that all three boys sit together using the formula:

\(\text{Probability} = \frac{\text{Number of favorable arrangements}}{\text{Total number of possible arrangements}}\)

\(\text{Probability} = \frac{144}{720}\)

We can simplify this fraction:

\(\frac{144}{720} = \frac{144 \div 144}{720 \div 144} = \frac{1}{5}\)

Alternatively, we can simplify step-by-step:

\(\frac{144}{720} = \frac{12 \times 12}{60 \times 12} = \frac{12}{60} = \frac{12 \div 12}{60 \div 12} = \frac{1}{5}\)

The probability that all three boys sit together is \(\frac{1}{5}\).

Summary of Calculations

Final Answer Determination

Based on our calculations, the probability that all three boys sit together is \(\frac{1}{5}\). This corresponds to Option 1.

Revision Table: Key Concepts in Probability & Arrangements

Additional Information: Related Probability Problems

Understanding how to calculate permutations and combinations is crucial for solving probability problems involving arrangements. Here are some related concepts and problem types:

• Probability of items NOT sitting together: Calculate the total arrangements and subtract the arrangements where they DO sit together.
• Probability of specific items at ends: Calculate arrangements where specific items are fixed at the ends, then arrange the rest.
• Probability with identical items: Use the formula for permutations with repetitions if some items are identical.
• Probability using Combinations: Use combinations when the order of selection doesn't matter, typically used when forming groups or committees.

These concepts build upon the fundamental principles of counting and probability, helping solve a wider range of arrangement problems.