
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

To solve this problem, we need to distribute 7 people into 3 vehicles such that all vehicles are used, and each vehicle can accommodate a maximum of 5 persons. The key here is that no vehicle should be empty.

### Step-by-step solution:

1. **Constraints on Vehicle Capacity:**
   Each vehicle can hold a maximum of 5 people. This limits us to configurations where the total number of people distributed among the vehicles still respects these constraints.

2. **Possible Groupings:**
   We need to find ways to divide 7 people into 3 non-empty groups. The number of people in each group (vehicle) must sum to 7. The possible distributions of people among the three vehicles (without exceeding the capacity of 5 for any vehicle) are:
   
   - \( (5, 1, 1) \)
   - \( (4, 2, 1) \)
   - \( (3, 3, 1) \)
   - \( (3, 2, 2) \)

3. **Counting the Ways to Assign People to Groups:**
   For each grouping, we calculate how many ways we can assign the people to the vehicles:

   - **For \( (5, 1, 1) \):**
     Choose 5 people out of 7 to go in the first vehicle. This can be done in \( \binom{7}{5} = \binom{7}{2} = 21 \) ways.
     After that, choose 1 person from the remaining 2 for the second vehicle. This can be done in \( \binom{2}{1} = 2 \) ways.
     The last person automatically goes to the third vehicle. So, the total number of ways is:
     \[
     21 \times 2 \times 1 = 42 \text{ ways}
     \]

   - **For \( (4, 2, 1) \):**
     Choose 4 people out of 7 to go in the first vehicle. This can be done in \( \binom{7}{4} = 35 \) ways.
     Then choose 2 people from the remaining 3 for the second vehicle. This can be done in \( \binom{3}{2} = 3 \) ways.
     The last person goes to the third vehicle. So, the total number of ways is:
     \[
     35 \times 3 \times 1 = 105 \text{ ways}
     \]

   - **For \( (3, 3, 1) \):**
     Choose 3 people out of 7 to go in the first vehicle. This can be done in \( \binom{7}{3} = 35 \) ways.
     Then choose 3 people from the remaining 4 for the second vehicle. This can be done in \( \binom{4}{3} = 4 \) ways.
     The last person goes to the third vehicle. So, the total number of ways is:
     \[
     35 \times 4 \times 1 = 140 \text{ ways}
     \]

   - **For \( (3, 2, 2) \):**
     Choose 3 people out of 7 to go in the first vehicle. This can be done in \( \binom{7}{3} = 35 \) ways.
     Then choose 2 people from the remaining 4 for the second vehicle. This can be done in \( \binom{4}{2} = 6 \) ways.
     The remaining 2 people will go into the third vehicle. So, the total number of ways is:
     \[
     35 \times 6 \times 1 = 210 \text{ ways}
     \]

4. **Summing the Total Number of Ways:**
   The total number of ways to assign 7 people to the 3 vehicles while using all vehicles is the sum of the possibilities for each configuration:
   \[
   42 + 105 + 140 + 210 = 497 \text{ ways}
   \]

### Final Answer:
There are **497 ways** for 7 people to visit using all 3 vehicles.

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Would you like more details on any step, or have any other questions?

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Here are 5 related questions:
1. How would the problem change if one vehicle had a capacity of only 4 people?
2. How many ways can the 7 people visit if the number of vehicles is increased to 4?
3. What if the vehicles can only hold 3 persons each?
4. How does the solution change if the people are assigned unique seat numbers in the vehicles?
5. Can this be generalized for \( n \) people and \( k \) vehicles with different capacities?

**Tip**: When solving combinatorics problems, always check for constraints like group size or capacity to avoid overcounting.

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 to distribute 7 people into 3 vehicles with a maximum capacity of 5 people each, using all vehicles. The solution involves combinatorics and binomial coefficients, providing a detailed breakdown of how to assign the people across vehicles. Ideal for students in Grades 10-12.

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