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. `(3!)^2`
B. `6!`
C. `5!`
D. none of these

Answer

To solve the problem of 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), we will break it down into steps.

### Step 1: Calculate the total number of seats
Each van has:
- 3 seats in the front
- 4 seats in the back

So, each van has a total of:
\[ 3 + 4 = 7 \text{ seats} \]

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

### Step 2: Determine the total number of people
We have:
- 3 girls
- 9 boys

Thus, the total number of people is:
\[ 3 + 9 = 12 \text{ people} \]

### Step 3: Calculate the number of ways to seat 12 people in 14 seats
Since there are 14 seats and only 12 people, we need to choose 12 seats from the 14 available. The number of ways to choose 12 seats from 14 is given by the combination formula:
\[ \binom{14}{12} = \binom{14}{2} = \frac{14 \times 13}{2 \times 1} = 91 \]

After choosing the seats, we can arrange the 12 people in those seats. The number of arrangements of 12 people is given by:
\[ 12! \]

Thus, the total number of arrangements is:
\[ \binom{14}{12} \times 12! = 91 \times 12! \]

### Step 4: Calculate the arrangements with 3 girls sitting together
Now, we need to find the number of arrangements where the 3 girls sit together in the back row on adjacent seats.

1. **Treat the 3 girls as a single unit**: When the 3 girls sit together, we can think of them as one "block". This block can be arranged in \(3!\) ways.
2. **Calculate the arrangement of the "block" and boys**: Now we have this "block" of girls and the 9 boys, making a total of 10 units to arrange (the block + 9 boys).
3. **Determine the seating arrangement**: The back row has 4 seats, and we need to place the block of girls in those 4 seats. The block can occupy any 3 adjacent seats in the back row. The possible arrangements for the block in the back row are:
- Block in seats 1, 2, 3
- Block in seats 2, 3, 4

Thus, there are 2 ways to position the block of girls.

4. **Calculate total arrangements**: For each arrangement of the block, we can arrange the remaining 9 boys in the remaining seats. The number of ways to arrange 9 boys in the remaining seats is \(9!\).

So, the total arrangements where the 3 girls sit together in the back row is:
\[ 2 \times 3! \times 9! \]

### Final Answer
The total number of arrangements is:
- Total arrangements of 12 people in 14 seats: \( 91 \times 12! \)
- Arrangements with 3 girls together: \( 2 \times 3! \times 9! \)