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 3 Vehicle 
a.3600
b.1500
c.1650
d.1806

This problem involves distributing 7 people among 3 vehicles, with the restriction that each vehicle can accommodate a maximum of 5 people and all 3 vehicles must be used. 

### Step-by-Step Solution:

#### Step 1: Set up the problem variables
- Let \( x_1, x_2, x_3 \) be the number of people in each vehicle.
- These variables must satisfy the equation:  
  \[
  x_1 + x_2 + x_3 = 7
  \]
  subject to the condition that \( 1 \leq x_1, x_2, x_3 \leq 5 \), because each vehicle must carry at least one person and at most five people.

#### Step 2: Solve for valid distributions
We need to find the number of integer solutions to the equation \( x_1 + x_2 + x_3 = 7 \) under the given constraints. Since all variables must be at least 1, we can perform a change of variables:
- Let \( y_1 = x_1 - 1 \), \( y_2 = x_2 - 1 \), and \( y_3 = x_3 - 1 \).  
  Now the equation becomes:  
  \[
  (y_1 + 1) + (y_2 + 1) + (y_3 + 1) = 7
  \]
  Simplifying:  
  \[
  y_1 + y_2 + y_3 = 4
  \]
  Now, \( y_1, y_2, y_3 \geq 0 \), and the maximum value for each \( x_i \) is 5, so \( y_i \leq 4 \). This is now a problem of finding the non-negative integer solutions to \( y_1 + y_2 + y_3 = 4 \) where each \( y_i \leq 4 \).

#### Step 3: Count the number of solutions
This is a stars and bars problem, but with restricted upper limits. We count the valid distributions of 4 stars among the three variables \( y_1, y_2, y_3 \). The possible valid distributions are:
- \( (4, 0, 0), (3, 1, 0), (2, 2, 0), (2, 1, 1) \), and permutations of these.

#### Step 4: Account for arrangements
Finally, for each valid distribution of people across the 3 vehicles, we need to account for the different ways the 7 people can be assigned to the vehicles. 

#### Step 5: Multiply by the possible arrangements


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 3 vehicles?
a. 3600
b. 1500
c. 1650
d. 1806

Learn how to solve the combinatorics problem of distributing 7 people across 3 vehicles, with each vehicle holding a maximum of 5 people. This problem explores the stars and bars method and restricted integer partitions. Suitable for Grades 11-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation