7 people visited a marriage function in 3 different vehicles, each accommodating a maximum of 5 persons. in how many ways can they set, sach that they use of the vehicles

Pick ONE option

3600

1500

1650

1806

We are given 7 people who need to be seated in 3 vehicles, with each vehicle accommodating a maximum of 5 persons. The goal is to find how many ways they can be seated in such a way that all vehicles are used.

### Step-by-step breakdown:

1. **Total number of people**: There are 7 people in total.
2. **Distribution of people across vehicles**: We must ensure that all 3 vehicles are used, so the possible distribution of people across the vehicles can only be \( (5, 1, 1) \). This is because each vehicle can hold up to 5 people, and the other two must have at least one person to satisfy the condition that all vehicles are used.

### Steps to solve:

- First, choose which vehicle will have 5 people. This can be done in \( \binom{3}{1} = 3 \) ways (since we are selecting 1 vehicle out of 3).
- Now, out of 7 people, choose 5 to go into the selected vehicle. This can be done in \( \binom{7}{5} = \binom{7}{2} = 21 \) ways.
- The remaining 2 people will be divided between the other two vehicles. Since each of these vehicles will get exactly 1 person, the number of ways to assign 1 person to each vehicle is \( 2! = 2 \).

### Total number of ways:
\[
\text{Total ways} = 3 \times 21 \times 2 = 126 \times 2 = 252
\]

Finally, the correct option provided is **not listed** based on these calculations.

The calculation might miss the specific options listed

7 people visited a marriage function in 3 different vehicles, each accommodating a maximum of 5 persons. In how many ways can they set, such that they use all of the vehicles? Options: 3600, 1500, 1650, 1806.

This problem involves seating 7 people in 3 vehicles where each vehicle can hold a maximum of 5 people. Explore how to solve this combinatorics problem using permutations and combinations, ensuring all vehicles are used. Suitable for students in Grades 9-12.

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