The number of ways in which three girls and ten boys can be seated in two vans, each having numbered seats, three in the front and four at the back is

Options

A. `""^(14)C_(12)`
B. `""^(14)P_(13)`
C. `14!`
D. `10^(3)`

Answer

To solve the problem of seating 3 girls and 10 boys in two vans, each having 3 seats in the front and 4 seats in the back, we can follow these steps:

### Step 1: Determine the total number of people and seats
We have a total of 3 girls and 10 boys, which gives us:
- Total people = 3 girls + 10 boys = 13 people
- Each van has 3 seats in the front and 4 seats in the back, so:
- Seats per van = 3 (front) + 4 (back) = 7 seats
- Total seats in both vans = 7 seats/van * 2 vans = 14 seats

### Step 2: Calculate the number of arrangements
We need to arrange 13 people in 14 seats. Since there are more seats than people, we can choose any 13 seats from the 14 available seats and arrange the 13 people in those seats.

The number of ways to choose 13 seats from 14 is given by the combination formula:
\[ \binom{14}{13} = 14 \]
After choosing the seats, we can arrange the 13 people in those seats. The number of arrangements of 13 people is given by:
\[ 13! \text{ (factorial of 13)} \]

### Step 3: Combine the calculations
The total number of ways to arrange the 3 girls and 10 boys in the two vans is:
\[ \text{Total arrangements} = \binom{14}{13} \times 13! = 14 \times 13! \]

### Final Calculation
Now, we can calculate \(13!\):
\[ 13! = 6227020800 \]
Thus, the total arrangements will be:
\[ 14 \times 6227020800 = 86978291200 \]

### Final Answer
The number of ways in which three girls and ten boys can be seated in two vans is **86,978,291,200**.

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