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?

This problem can be solved by considering the constraints: there are 7 people, 3 vehicles, and each vehicle can carry a maximum of 5 people. We need to find the number of ways to distribute the 7 people among the 3 vehicles such that all vehicles are used.

### Step-by-step Solution:

1. **Distribution of People in Vehicles:**
   - We need to distribute 7 people into 3 vehicles such that all vehicles are used. Let's denote the number of people in each vehicle as \(x_1\), \(x_2\), and \(x_3\) where \(x_1 + x_2 + x_3 = 7\), and each \(x_i \geq 1\) because all vehicles must be used.
   
   - The problem is to find how many ways we can assign the values to \(x_1\), \(x_2\), and \(x_3\), satisfying \(x_1 + x_2 + x_3 = 7\), where each \(x_i\) is at least 1, and each \(x_i \leq 5\), since each vehicle has a maximum capacity of 5 people.

2. **Transforming the Problem:**
   - Let’s define new variables \(y_1 = x_1 - 1\), \(y_2 = x_2 - 1\), and \(y_3 = x_3 - 1\), to ensure each vehicle gets at least 1 person.
   - Now, we need to solve \(y_1 + y_2 + y_3 = 4\), where \(y_1, y_2, y_3 \geq 0\) and \(y_1 + 1 \leq 5\), i.e., \(y_1 \leq 4\), and similarly for \(y_2\) and \(y_3\).

3. **Counting the Number of Solutions:**
   - The equation \(y_1 + y_2 + y_3 = 4\) is a classic "stars and bars" problem without any restrictions, and the number of non-negative integer solutions is given by the combination formula:
     \[
     \binom{4 + 2}{2} = \binom{6}{2} = 15
     \]
   
4. **Distributing the People Among the Vehicles:**
   - After determining how many people go into each vehicle, we need to assign specific people to those vehicles. The number of ways to assign 7 people to the vehicles based on the distribution is given by the multinomial coefficient:
     \[
     \frac{7!}{x_1! x_2! x_3!}
     \]
   - Multiply the two counts together to get the total number of possible distributions.

Thus,

Explore the combinatorial problem of distributing 7 people into 3 vehicles, with a maximum capacity of 5 persons per vehicle. Learn how to apply the Stars and Bars method and multinomial coefficients to solve this problem, suitable for students in Grades 9-11.

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