7 person visited a marriage function in 3 different vehicle each accomodating a maximum of 5 persons.in how many can they visit so they use all three vehicle 3600 1500 1650 1806

We are given that 7 people visited a marriage function in 3 different vehicles, each accommodating a maximum of 5 persons. The goal is to find how many ways they can distribute the people across the vehicles such that all three vehicles are used.

### Step-by-step approach:

1. **Understand the distribution problem**:
   - We have 7 people to distribute among 3 vehicles.
   - Each vehicle can hold up to 5 people.
   - All three vehicles must be used.

2. **Find the valid distribution**:
   - Since the vehicles can hold a maximum of 5 people each and there are only 7 people, one possible distribution is (3, 2, 2):
     - One vehicle carries 3 people.
     - The other two vehicles carry 2 people each.

3. **Number of ways to assign people**:
   - First, we choose 3 people out of the 7 for the vehicle that will carry 3 passengers. This can be done in \( \binom{7}{3} \) ways.
   - Then, we choose 2 people from the remaining 4 for the second vehicle. This can be done in \( \binom{4}{2} \) ways.
   - The last 2 people automatically go to the third vehicle, so there is only 1 way to assign them.

4. **Account for vehicle arrangements**:
   - Since the three vehicles are distinct, the different arrangements of the vehicles must be taken into account. There are \( 3! = 6 \) ways to assign the three groups to the three vehicles.

5. **Calculate the total number of ways**:
   - The total number of ways is the product of all these choices:

\[
\binom{7}{3} \times \binom{4}{2} \times 1 \times 3!
\]

Now let's compute the result.The total number of ways the 7 people can visit the marriage function using all three vehicles is **1260**.

Would you like further details on any part of the solution? Here are some related questions:

1. How can we generalize this problem for any number of people and vehicles?
2. What if one vehicle can hold fewer than 5 people?
3. How would the solution change if the vehicles were identical?
4. How do binomial coefficients work in problems like this?
5. How does the use of factorials apply in different arrangement problems?

**Tip:** When working with distribution problems, always check if there's a restriction on using all items (vehicles, groups, etc.), as it significantly changes the calculations.

7 person visited a marriage function in 3 different vehicle each accommodating a maximum of 5 persons. In how many ways can they visit so they use all three vehicles?

Explore how to distribute 7 people across 3 vehicles, each accommodating a maximum of 5 persons, in such a way that all three vehicles are used. The solution involves combinatorics concepts such as binomial coefficients and factorials. Suitable for students in Grades 10-12.

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