This problem involves distributing 7 people among 3 vehicles, with the restriction that each vehicle can accommodate a maximum of 5 people and all 3 vehicles must be used.

Step-by-Step Solution:

Step 1: Set up the problem variables

• Let $x_1, x_2, x_3$ be the number of people in each vehicle.
• These variables must satisfy the equation:
  $x_1 + x_2 + x_3 = 7$ subject to the condition that $1 \leq x_1, x_2, x_3 \leq 5$, because each vehicle must carry at least one person and at most five people.

Step 2: Solve for valid distributions

We need to find the number of integer solutions to the equation $x_1 + x_2 + x_3 = 7$ under the given constraints. Since all variables must be at least 1, we can perform a change of variables:

• Let $y_1 = x_1 - 1$, $y_2 = x_2 - 1$, and $y_3 = x_3 - 1$.
  Now the equation becomes:
  $(y_1 + 1) + (y_2 + 1) + (y_3 + 1) = 7$ Simplifying:
  $y_1 + y_2 + y_3 = 4$ Now, $y_1, y_2, y_3 \geq 0$, and the maximum value for each $x_i$ is 5, so $y_i \leq 4$. This is now a problem of finding the non-negative integer solutions to $y_1 + y_2 + y_3 = 4$ where each $y_i \leq 4$.

Step 3: Count the number of solutions

This is a stars and bars problem, but with restricted upper limits. We count the valid distributions of 4 stars among the three variables $y_1, y_2, y_3$. The possible valid distributions are:

• $(4, 0, 0), (3, 1, 0), (2, 2, 0), (2, 1, 1)$, and permutations of these.

Step 4: Account for arrangements

Finally, for each valid distribution of people across the 3 vehicles, we need to account for the different ways the 7 people can be assigned to the vehicles.

Step 5: Multiply by the possible arrangements