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?

Options

A. `""^(14)c_(12)`
B. `9^4`
C. `""^(14)p_(12)`
D. none of these

Answer

To solve the problem of seating 3 girls and 9 boys in two vans, we need to determine the number of ways to arrange 12 people in 14 seats (3 in the front and 4 at the back of each van). Here’s a step-by-step breakdown of the solution:

### Step 1: Determine the Total Number of Seats
Each van has:
- 3 seats in the front
- 4 seats in the back

Since there are 2 vans, the total number of seats is:
\[ \text{Total Seats} = 2 \times (3 + 4) = 2 \times 7 = 14 \]

### Step 2: Determine the Total Number of People
We have:
- 3 girls
- 9 boys

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

### Step 3: Calculate the Number of Ways to Seat the People
Since we have 14 seats and 12 people, we can choose any 12 seats from the 14 available. The number of ways to arrange 12 people in 14 seats can be calculated using permutations, as the order in which they are seated matters.

The formula for permutations is given by:
\[ nPr = \frac{n!}{(n-r)!} \]
where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

In this case, we need to calculate:
\[ 14P12 = \frac{14!}{(14-12)!} = \frac{14!}{2!} \]

### Step 4: Simplify the Calculation
Calculating \( 14! \) and \( 2! \):
- \( 14! = 14 \times 13 \times 12! \)
- \( 2! = 2 \times 1 = 2 \)

Thus,
\[ 14P12 = \frac{14 \times 13 \times 12!}{2} = 14 \times 13 \times 12! / 2 \]

### Step 5: Final Calculation
Now we can compute the value:
\[ 14 \times 13 = 182 \]
So,
\[ 14P12 = \frac{182 \times 12!}{2} = 91 \times 12! \]

### Conclusion
The total number of ways to seat 3 girls and 9 boys in the two vans is:
\[ 14P12 = 91 \times 12! \]