If 8 boys and 6 girls sit in a row, then probability that all girls are together is

Options

A. `3/999`
B. `3/1000`
C. `3/1001`
D. `3/10`

Answer

To find the probability that all girls are sitting together when 8 boys and 6 girls are seated in a row, we can follow these steps:

### Step 1: Calculate the total arrangements of 14 people
First, we need to find the total number of ways to arrange 8 boys and 6 girls in a row. Since there are 14 people in total (8 boys + 6 girls), the total arrangements can be calculated using the factorial of the total number of people:

\[ \text{Total arrangements} = 14! \]

### Step 2: Treat the group of girls as a single unit
To find the favorable arrangements where all girls are together, we can treat the 6 girls as one single unit or bundle. Thus, we have:

- 1 bundle of girls
- 8 boys

This gives us a total of 9 units to arrange (8 boys + 1 bundle of girls).

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

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

### Step 4: Calculate the arrangements of girls within their bundle
Within the bundle of girls, the 6 girls can be arranged among themselves in:

\[ \text{Arrangements of girls} = 6! \]

### Step 5: Calculate the favorable arrangements
The total number of favorable arrangements where all girls are together is the product of the arrangements of the units and the arrangements of the girls within their bundle:

\[ \text{Favorable arrangements} = 9! \times 6! \]

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

\[ P(\text{all girls together}) = \frac{\text{Favorable arrangements}}{\text{Total arrangements}} = \frac{9! \times 6!}{14!} \]

### Step 7: Simplify the probability
Now we can simplify the probability:

\[ P(\text{all girls together}) = \frac{9! \times 6!}{14!} = \frac{9! \times 6!}{14 \times 13 \times 12 \times 11 \times 10 \times 9!} \]

The \(9!\) in the numerator and denominator cancels out:

\[ P(\text{all girls together}) = \frac{6!}{14 \times 13 \times 12 \times 11 \times 10} \]

Calculating \(6!\):

\[ 6! = 720 \]

Now substituting that back in:

\[ P(\text{all girls together}) = \frac{720}{14 \times 13 \times 12 \times 11 \times 10} \]

Calculating the denominator:

\[ 14 \times 13 = 182 \]
\[ 182 \times 12 = 2184 \]
\[ 2184 \times 11 = 24024 \]
\[ 24024 \times 10 = 240240 \]

So, we have:

\[ P(\text{all girls together}) = \frac{720}{240240} \]

Now simplifying this fraction:

\[ P(\text{all girls together}) = \frac{3}{1001} \]

Thus, the final answer is:

\[ \text{Probability that all girls are together} = \frac{3}{1001} \]