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!