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 whicdes?

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3600

1500

1650

1806

We are given that 7 people are visiting a function using 3 vehicles, and each vehicle can accommodate a maximum of 5 persons. We need to find the number of ways they can visit, ensuring that all 3 vehicles are used.

### Step-by-step breakdown:

1. **Partition the 7 people into 3 groups** (since all 3 vehicles must be used):
   - Since each vehicle can hold a maximum of 5 people, we need to split the group of 7 into 3 parts such that no part exceeds 5. The valid partition is \( (3, 2, 2) \). This means one vehicle will carry 3 people, and the other two will carry 2 people each.

2. **Choose who goes in each vehicle**:
   - First, select 3 people out of 7 to go into the vehicle with 3 people. The number of ways to do this is \( \binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \).
   - Then, from the remaining 4 people, select 2 people to go into one of the vehicles with 2 people. This can be done in \( \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6 \).
   - The remaining 2 people automatically go into the last vehicle, and there is only 1 way to assign them to this vehicle.

3. **Account for the arrangement of the vehicles**:
   - Since the vehicles are distinct, the 3 groups can be assigned to the vehicles in \( 3! = 6 \) ways.

### Final Calculation:
The total number of ways is given by:
\[
\binom{7}{3} \times \binom{4}{2} \times 1 \times 3! = 35 \times 6 \times 6 = 1260
\]

It seems the answer 1260 is not an option in the list. Upon reviewing, it seems there's an assumption difference or an overlooked step.

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?

Explore a combinatorics problem involving the partitioning of 7 people into 3 distinct vehicles, each holding up to 5 people. This problem covers key concepts such as combinations, permutations, and factorials. Detailed step-by-step solution provided for students in Grades 11-12.

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