Math Problem Statement

7 people visited a marriage function in 3 different vehicles, each accommodating a maximum of 5 persons. in how many ways can they set, sach that they use of the vehicles

Pick ONE option

3600

1500

1650

1806

Solution

We are given 7 people who need to be seated in 3 vehicles, with each vehicle accommodating a maximum of 5 persons. The goal is to find how many ways they can be seated in such a way that all vehicles are used.

Step-by-step breakdown:

  1. Total number of people: There are 7 people in total.
  2. Distribution of people across vehicles: We must ensure that all 3 vehicles are used, so the possible distribution of people across the vehicles can only be (5,1,1)(5, 1, 1). This is because each vehicle can hold up to 5 people, and the other two must have at least one person to satisfy the condition that all vehicles are used.

Steps to solve:

  • First, choose which vehicle will have 5 people. This can be done in (31)=3\binom{3}{1} = 3 ways (since we are selecting 1 vehicle out of 3).
  • Now, out of 7 people, choose 5 to go into the selected vehicle. This can be done in (75)=(72)=21\binom{7}{5} = \binom{7}{2} = 21 ways.
  • The remaining 2 people will be divided between the other two vehicles. Since each of these vehicles will get exactly 1 person, the number of ways to assign 1 person to each vehicle is 2!=22! = 2.

Total number of ways:

Total ways=3×21×2=126×2=252\text{Total ways} = 3 \times 21 \times 2 = 126 \times 2 = 252

Finally, the correct option provided is not listed based on these calculations.

The calculation might miss the specific options listed

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Math Problem Analysis

Mathematical Concepts

Combinatorics
Permutations
Combinations

Formulas

Combination formula: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)
Factorial formula: n! = n × (n-1) × ... × 1

Theorems

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Suitable Grade Level

Grades 9-12