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?

This problem involves distributing people across 3 vehicles, ensuring all vehicles are used, with a maximum of 5 persons per vehicle.

Let’s break it down:

1. **Total number of people**: Each vehicle can accommodate a maximum of 5 people, so in total, there are \( 3 \times 5 = 15 \) people.
   
2. **Vehicles usage condition**: Since we must use all 3 vehicles, we need to divide the 15 people into 3 groups in a way that ensures each vehicle is occupied by at least one person.

This is a classic problem of partitioning with the restriction that no group is empty. The number of ways to distribute 15 people into 3 groups such that all groups are non-empty is given by **the Stirling number of the second kind** for 3 groups, and then multiplying by the number of ways to arrange the groups across the vehicles.

However, since each group can have up to 5 people, we need to find a way to distribute exactly 5 people into each vehicle, and then multiply by the number of arrangements for vehicles.

**Steps**:
Let me clarify

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?

Learn how to distribute 15 people across 3 vehicles, each holding up to 5 people, ensuring all vehicles are used. This problem involves combinatorics, partitioning using Stirling numbers, and permutations. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation