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

We need to find the number of ways in which 7 people can visit a marriage function using 3 different vehicles, each accommodating a maximum of 5 persons, such that all the vehicles are used.

### Step-by-step explanation:

1. **Divide 7 people into 3 groups** such that all vehicles are used, meaning none of the groups can be empty. Since each vehicle has a capacity of 5, the possible group sizes must fit within the constraint of 7 people total.

The possible group sizes for the vehicles are:
- 5 people in one vehicle, 1 person in another, and 1 person in the third.
- 4 people in one vehicle, 2 people in another, and 1 person in the third.

We will calculate the number of ways for each of these cases.

### Case 1: Groups of (5, 1, 1)
- Choose 5 people to go in the first vehicle:  
  \[
  \binom{7}{5} = \binom{7}{2} = 21
  \]
- Choose 1 person for the second vehicle from the remaining 2 people:  
  \[
  \binom{2}{1} = 2
  \]
- The remaining 1 person will go in the third vehicle.  
  There's only 1 way to assign this person.

Thus, the total number of ways for this case is:
\[
21 \times 2 \times 1 = 42 \text{ ways}
\]

### Case 2: Groups of (4, 2, 1)
- Choose 4 people to go in the first vehicle:  
  \[
  \binom{7}{4} = \binom{7}{3} = 35
  \]
- Choose 2 people for the second vehicle from the remaining 3 people:  
  \[
  \binom{3}{2} = 3
  \]
- The remaining 1 person will go in the third vehicle.  
  There's only 1 way to assign this person.

Thus, the total number of ways for this case is:
\[
35 \times 3 \times 1 = 105 \text{ ways}
\]

### Total number of ways
Now, we add the results from both cases:
\[
42 + 105 = 147
\]

Therefore, the total number of ways in which 7 people can visit using all 3 vehicles is **147 ways**.

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Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would the problem change if one vehicle could carry more than 5 people?
2. How many ways would there be if the vehicles had no capacity limit?
3. What if only two vehicles had to be used?
4. How does the solution change if the people are arranged in a specific seating order within each vehicle?
5. How does this combinatorics problem relate to permutations?

**Tip:** When breaking problems into cases, ensure all scenarios are covered, including boundary conditions.

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 how 7 people can be divided into groups to fit in 3 vehicles with seating constraints. Using combinations and permutations, this problem provides a step-by-step solution involving groups of (5,1,1) and (4,2,1). Suitable for students in Grades 10-12.

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