7 people visited a marriage function in 3 different vehicles each accomodating max of 5 ppl how many ways can they visit such that they use all 3 vehicles 

We are asked to find how many ways 7 people can visit a marriage function using 3 vehicles, each with a capacity of 5 people, such that all 3 vehicles are used.

### Step 1: Partitioning the 7 people into 3 groups
We need to split 7 people into 3 groups such that none of the groups is empty (since all 3 vehicles must be used). The groups can have the following distribution:
- One group with 3 people, one group with 2 people, and one group with 2 people.
  
### Step 2: Counting the number of ways to partition the people
We calculate the number of ways to assign people to these groups.

1. **Choose 3 people for the first group**: There are \( \binom{7}{3} \) ways to choose 3 people.
   
   \[
   \binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35
   \]

2. **Choose 2 people for the second group**: After choosing 3 people for the first group, we are left with 4 people. There are \( \binom{4}{2} \) ways to choose 2 people for the second group.
   
   \[
   \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6
   \]

3. **The remaining 2 people automatically go to the third group**: There's only 1 way to assign these last 2 people.

### Step 3: Considering the arrangement of vehicles
Since the vehicles are distinct, the arrangement of the groups into the vehicles matters. There are 3! (factorial) ways to assign the 3 groups to the 3 vehicles:

\[
3! = 6
\]

### Step 4: Final calculation
Now, multiply all the factors together:

\[
\text{Total ways} = \binom{7}{3} \times \binom{4}{2} \times 3! = 35 \times 6 \times 6 = 1260
\]

Thus, the total number of ways the 7 people can visit using all 3 vehicles is **1260**.

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Would you like more details on any of the steps?

### Related Questions:
1. How do you calculate the binomial coefficient \(\binom{n}{r}\)?
2. What is the principle behind arranging groups into distinct sets?
3. How would the answer change if one vehicle could hold only 3 people?
4. Can you solve similar problems where some vehicles have larger capacities?
5. How does Stirling’s formula approximate these types of combinatorial problems?

### Tip:
When solving combinatorics problems, breaking the task into smaller, manageable steps—like grouping and arranging—is key to avoiding mistakes!

7 people visited a marriage function in 3 different vehicles each accommodating max of 5 people. How many ways can they visit such that they use all 3 vehicles?

Discover how to solve a combinatorics problem involving 7 people visiting a marriage function in 3 vehicles. Learn the step-by-step process of partitioning people into groups, applying binomial coefficients, and calculating factorials to find that there are 1260 ways to assign the groups to vehicles. Ideal for students in Grades 9-12.

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