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

The problem is asking for the number of ways to arrange 7 people in 3 vehicles, where each vehicle can accommodate a maximum of 5 people, and all vehicles must be used.

We can solve this using the following steps:

### Step 1: Distribute 7 people into 3 vehicles
Let the number of people in each vehicle be \( x_1 \), \( x_2 \), and \( x_3 \), where \( x_1 + x_2 + x_3 = 7 \) and each vehicle has at least 1 person (since all 3 vehicles are used). Additionally, each vehicle can have a maximum of 5 people. So, we need to find the number of integer solutions to:

\[
x_1 + x_2 + x_3 = 7
\]
where \( 1 \leq x_1, x_2, x_3 \leq 5 \).

To handle this, let's first transform the variables by letting \( y_1 = x_1 - 1 \), \( y_2 = x_2 - 1 \), and \( y_3 = x_3 - 1 \), so that now:

\[
y_1 + y_2 + y_3 = 4
\]
where \( 0 \leq y_1, y_2, y_3 \leq 4 \).

### Step 2: Count the number of solutions
This is now a stars and bars problem, where we need to find the number of non-negative integer solutions to \( y_1 + y_2 + y_3 = 4 \), subject to the constraint that each \( y_i \leq 4 \).

The number of unrestricted solutions (without the upper limit of 4) is given by the stars and bars formula:

\[
\binom{4+2}{2} = \binom{6}{2} = 15.
\]

Since none of the variables can exceed 4, and this is already within the allowable range, no further adjustments are necessary.

### Step 3: Permutations within the groups
Now that we have determined the number of ways to distribute people among the vehicles, we need to account for the number of ways to assign specific people to the vehicles.

After choosing how many people go in each vehicle, we can arrange the 7 people in those vehicles. This is done by permuting the 7 people across the groups. So the number of ways to assign specific people to the groups is:

\[
\frac{7!}{x_1!x_2!x_3!}
\]

However, the details of this require further clarification depending on specific numbers of

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 3 vehicles?

Discover how to arrange 7 people in 3 vehicles, each accommodating up to 5 people, using combinatorics and permutation methods. This problem involves the stars and bars theorem, as well as key permutation concepts. Suitable for students in Grades 11-12.

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