In how many ways three girls and nine boys can be seated in two vans, each having numbered seats, 3 in the front and 4 at the back? How many seating arrangements are possible if 3 girls sit together in a back row on adjacent seats?

Options

A. 12!
B. `2xx12!`
C. `2xx10!`
D. `2xx9!`

Answer

To solve the problem of seating arrangements for three girls and nine boys in two vans, we need to break down the problem step by step.

### Step 1: Understand the seating arrangement in the vans
Each van has:
- 3 seats in the front
- 4 seats in the back

Thus, each van has a total of 7 seats. Since there are 2 vans, the total number of seats available is:
\[ 2 \times 7 = 14 \text{ seats} \]

### Step 2: Determine the arrangement when 3 girls sit together
Since the problem states that the 3 girls must sit together in the back row on adjacent seats, we can treat the 3 girls as a single unit or block. This block can occupy any 3 adjacent seats in the back row of either van.

### Step 3: Arrangements of the girls
The girls can be arranged among themselves in the block in:
\[ 3! = 6 \text{ ways} \]
Additionally, since the block can be placed in either of the two vans, we have:
\[ 2 \text{ (vans)} \]

### Step 4: Determine the arrangement of the remaining boys
After seating the 3 girls, we have:
- Total people = 12 (3 girls + 9 boys)
- Remaining seats = 14 - 3 = 11 seats

Now, we need to arrange the remaining 9 boys in the 11 available seats. The number of ways to choose 9 seats from 11 is given by the combination formula \( \binom{n}{r} \):
\[ \binom{11}{9} = \binom{11}{2} = \frac{11 \times 10}{2 \times 1} = 55 \]

### Step 5: Arrangements of the boys
Once the seats are chosen, the boys can be arranged in those seats in:
\[ 9! \text{ ways} \]

### Step 6: Combine all arrangements
Now, we can combine all the arrangements:
- Arrangements of the girls: \( 6 \) ways
- Choices of seats for the boys: \( 55 \) ways
- Arrangements of the boys: \( 9! \) ways
- Choices of vans: \( 2 \) ways

Thus, the total number of seating arrangements is:
\[ \text{Total arrangements} = 6 \times 2 \times 55 \times 9! \]

### Step 7: Calculate the total arrangements
Now, substituting \( 9! = 362880 \):
\[ \text{Total arrangements} = 6 \times 2 \times 55 \times 362880 \]
Calculating this step-by-step:
1. \( 6 \times 2 = 12 \)
2. \( 12 \times 55 = 660 \)
3. \( 660 \times 362880 = 239500800 \)

Thus, the total number of seating arrangements is:
\[ \text{Total arrangements} = 239500800 \]

### Final Answer
The total number of ways to seat three girls and nine boys in the two vans, with the condition that the girls sit together, is **239500800**.