7 people visited a marriage function in 3 different vehicles, each accommodating a maximum of 5 persons. In how many ways can then visit, such that they use all the 3 vehicles?
Pick ONE option
3600
1500
1650
1806
Clear Selection

We are given a problem where 7 people need to visit a marriage function using 3 different vehicles, each of which can accommodate a maximum of 5 persons. We are required to find how many ways the people can visit such that all 3 vehicles are used.

### Approach:
- First, we need to distribute 7 people into 3 vehicles, and since all 3 vehicles must be used, no vehicle can be empty. 
- This means we need to find how to partition 7 people into three non-empty groups, and the group sizes are restricted by the vehicle capacity (max 5 per vehicle).

The possible way to partition 7 people into 3 non-empty groups is:
- One vehicle with 3 people, and two vehicles with 2 people each (since any other partition violates the vehicle capacity).

### Steps:
1. **Choose the groups**: 
   - First, choose 3 people to go in the vehicle with 3 seats. The number of ways to do this is \(\binom{7}{3}\).
   - After that, choose 2 people to go into the first vehicle with 2 seats. The number of ways to do this is \(\binom{4}{2}\).
   - The remaining 2 people automatically go into the second vehicle with 2 seats.

2. **Permute the vehicles**: 
   - There are 3 vehicles, and any of them can be assigned to one of the groups. Since the vehicles are distinct, we need to account for the permutation of the 3 groups among the 3 vehicles. This can be done in \(3!\) ways.

### Calculations:
- \(\binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35\)
- \(\binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6\)
- \(3! = 6\)

Now, multiplying all these together:
\[
35 \times 6 \times 6 = 1260
\]

Thus, the total number of ways is **1260**. Since this option is not available, let's verify the restrictions once more.

Considering that I may have misinterpreted the number

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? Pick ONE option: 3600, 1500, 1650, 1806.

Solve a combinatorics problem where 7 people need to be distributed across 3 vehicles with a maximum capacity of 5 people per vehicle. Learn how to use combinations and permutations to find the total number of ways this can be done. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation