7 people visited a marriage function in 3 different vehicles, each accommodating a maximum of 5 persons. In how many ways can they visit, such that they use all the 3 vehicles?

To solve this problem, we are tasked with determining the number of ways 7 people can travel using 3 vehicles, where each vehicle can hold a maximum of 5 persons. Additionally, all 3 vehicles must be used.

### Step-by-Step Solution:

1. **Basic Setup**:
   - There are 7 people in total.
   - There are 3 vehicles.
   - Each vehicle has a maximum capacity of 5 persons.
   - We are asked to find how many ways these 7 people can be assigned to the vehicles, ensuring that all 3 vehicles are used.

2. **Distribution of People Among Vehicles**:
   - Since all vehicles must be used, and there are 7 people in total, we need to find a valid distribution of people across the vehicles such that no vehicle is empty and the number of people assigned to each vehicle does not exceed 5.
   - We can distribute the people in the following ways (as each vehicle must contain at least 1 person):
     - (3, 2, 2): One vehicle carries 3 people, and the other two carry 2 people each.

3. **Calculation of Number of Ways to Distribute People**:
   - First, choose which vehicle will carry 3 people. This can be done in \(\binom{7}{3}\) ways.
   - After selecting 3 people for one vehicle, we are left with 4 people. Now, we need to assign 2 people to one of the two remaining vehicles. This can be done in \(\binom{4}{2}\) ways.
   - The remaining 2 people will automatically go to the last vehicle (no choice is needed here).

   So, the number of ways to distribute the people is:
   \[
   \binom{7}{3} \times \binom{4}{2} = \frac{7!}{3!(7-3)!} \times \frac{4!}{2!(4-2)!} = 35 \times 6 = 210
   \]

4. **Arrangements of Vehicles**:
   - Since the 3 vehicles are distinguishable, there are \(3!\) (i.e., 6) ways to assign these groups of people to the vehicles.

5. **Total Number of Ways**:
   - Finally, multiply the number of ways to distribute the people by the number of ways to arrange the vehicles:
   \[
   210 \times 6 = 1260
   \]

Thus, the total number of ways in which the 7 people can visit using all 3 vehicles is \( \boxed{1260} \).

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Would you like further details on any part of this solution?

Here are 5 related questions:
1. What if one vehicle could accommodate a maximum of 6 people?
2. How would the solution change if one vehicle had a capacity of only 4 people?
3. Can you generalize this solution for \( n \) people and \( k \) vehicles?
4. How would the solution change if the vehicles were not distinguishable?
5. What would happen if the vehicles didn't have a capacity limit?

**Tip:** When tackling combinatorial problems, first ensure you correctly understand the constraints (like capacity limits) and then focus on how to distribute items or people logically.

Explore how to solve the combinatorics problem of distributing 7 people into 3 vehicles, each with a maximum capacity of 5, ensuring all vehicles are used. This solution involves calculating permutations and combinations, with a step-by-step breakdown of key formulas and concepts. Suitable for high school students in Grades 9-12.

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