If 8 boys and 2 girls are to be seated in a row for a photograph, find the probability that the girls sit together.

Answer

To find the probability that the two girls sit together when 8 boys and 2 girls are seated in a row for a photograph, we can follow these steps:

### Step 1: Calculate the total number of arrangements of 10 students
The total number of students is 8 boys + 2 girls = 10 students. The total arrangements of these 10 students can be calculated using the factorial notation:

\[ \text{Total arrangements} = 10! \]

### Step 2: Treat the two girls as a single unit
To find the arrangements where the girls sit together, we can treat the two girls as a single entity or unit. This means we now have:

- 8 boys
- 1 unit (which consists of the 2 girls)

So, we have a total of 8 boys + 1 unit = 9 units to arrange.

### Step 3: Calculate the arrangements of the 9 units
The number of ways to arrange these 9 units (8 boys + 1 unit of girls) is given by:

\[ \text{Arrangements of 9 units} = 9! \]

### Step 4: Arrange the girls within their unit
Within their unit, the two girls can be arranged among themselves in:

\[ \text{Arrangements of girls} = 2! \]

### Step 5: Calculate the number of favorable arrangements
The total number of favorable arrangements where the girls sit together is the product of the arrangements of the 9 units and the arrangements of the girls within their unit:

\[ \text{Favorable arrangements} = 9! \times 2! \]

### Step 6: Calculate the probability
The probability that the girls sit together is given by the ratio of the number of favorable arrangements to the total arrangements:

\[ P(\text{girls sit together}) = \frac{\text{Favorable arrangements}}{\text{Total arrangements}} = \frac{9! \times 2!}{10!} \]

### Step 7: Simplify the expression
We know that \(10! = 10 \times 9!\). Thus, we can simplify the probability:

\[ P(\text{girls sit together}) = \frac{9! \times 2!}{10 \times 9!} = \frac{2!}{10} = \frac{2}{10} = \frac{1}{5} \]

### Final Answer
The probability that the girls sit together is:

\[ \frac{1}{5} \]