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We are given a group of **12 people**, with the following traveling arrangements:

- **1 small car** can carry 1 driver and 1 passenger.
- **2 larger cars**, each of which can carry 1 driver and 4 passengers.
- There are **5 drivers** in the group, and we need to determine how many different ways they can travel.

### Step-by-step approach:

1. **Assigning the drivers:**
   - We need to assign **3 drivers** from the group of 5 drivers (1 driver for the small car and 2 drivers for the large cars).
   - The number of ways to choose 3 drivers from 5 can be calculated using combinations:
     \[
     \binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10
     \]
   So, there are 10 ways to choose 3 drivers.

2. **Assigning passengers:**
   - After assigning the drivers, there are **9 remaining people** (since 3 drivers are already selected). We need to assign:
     - **1 passenger** to the small car.
     - **4 passengers** to the first large car.
     - **4 passengers** to the second large car.
   - The number of ways to choose 1 passenger for the small car from 9 people is:
     \[
     \binom{9}{1} = 9
     \]
   - After that, we have 8 people left. The number of ways to choose 4 passengers for the first large car from the remaining 8 people is:
     \[
     \binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70
     \]
   - The remaining 4 people must go into the second large car, and there is only 1 way to assign them.

3. **Total number of ways:**
   The total number of ways they can travel is the product of all these combinations:
   \[
   10 \times 9 \times 70 \times 1 = 6300
   \]

Thus, the group can travel in **6,300 different ways**.

Would you like more details or have any questions? 

Here are five related questions you might find helpful:
1. How do combinations work in counting problems?
2. How can this problem change if there are fewer drivers available?
3. What if the number of passengers per car changed?
4. How would adding more cars change the calculation?
5. How does permutation differ from combination in these problems?

**Tip:** Always consider splitting tasks into smaller steps and focus on one part at a time when solving complex counting problems.

A group of 12 people want to go to a concert. They can travel in a small car that takes one driver and one passenger and two cars each taking one driver and four passengers. If there are five drivers in the group, in how many different ways can they travel?

Learn how to solve a combinatorics problem involving 12 people traveling in different car arrangements with 5 available drivers. This problem uses combination formulas to determine the total number of ways the group can travel. Suitable for Grades 10-12.

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