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To solve the problem, we need to find the number of ways to allocate 12 people into two vehicles given the constraints that only two of the 12 people can drive and that the car can carry 4 people while the SUV can carry 8 people.

Here's a step-by-step breakdown of how to approach the problem:

1. **Identify the Drivers**: We have two drivers, let's call them Driver A and Driver B. Each vehicle needs one driver.

2. **Assign Drivers to Vehicles**:
    - Driver A can drive the car, and Driver B can drive the SUV.
    - Alternatively, Driver B can drive the car, and Driver A can drive the SUV.

    This gives us 2 ways to assign the drivers to the vehicles.

3. **Distribute the Remaining Passengers**:
    - After assigning the drivers, we have 10 people left to allocate (since 2 of the 12 are drivers).
    - The car can carry 3 more people (since the driver is already counted), and the SUV can carry 7 more people.

    We need to choose 3 out of the 10 remaining people to fill the car (the rest will go into the SUV).

    The number of ways to choose 3 people from 10 is given by the combination formula \( \binom{n}{k} \), where \( n \) is the total number of items to choose from, and \( k \) is the number of items to choose.

    \[
    \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10!}{3! \cdot 7!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120
    \]

4. **Calculate Total Number of Ways**:
    - We have 2 ways to assign the drivers.
    - We have 120 ways to choose 3 passengers out of the remaining 10 people to fill the car.

    The total number of ways to allocate all 12 people to the two vehicles is:

    \[
    2 \times 120 = 240
    \]

Therefore, there are 240 ways to allocate the 12 people to the two vehicles.

Would you like more details or have any questions? Here are some related questions for further exploration:

1. How would the solution change if there were more than two drivers available?
2. What if the car could carry 5 people instead of 4? How would that affect the number of ways to allocate the passengers?
3. If there were only one driver available, how many ways could the allocation be done?
4. How can we generalize this solution to any number of people and vehicles with different capacities?
5. What would the solution look like if the vehicles had the same capacity?

**Tip:** When dealing with combinations and permutations, it's helpful to clearly identify constraints and break the problem down into manageable steps.

12 people need to travel to a hockey match in two vehicles: a car, which can carry four people, and an SUV, which can carry eight people. Given that only two of the 12 can drive, find the number of ways they can be allocated to the two vehicles.

Learn how to allocate 12 people to two vehicles, where only two of them can drive, using combinations and basic counting principles. Solve this problem using key combinatorics concepts, suitable for high school students in grades 9-12.

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