Eight persons are to be transported from city A to city B in three cars of different makes. If each

Q: Why can't we just use the stars and bars method here?

Stars and bars is perfect for identical items, but here, the people are distinct individuals! If Person A is in Car 1, it is a completely different scenario than if Person B is in Car 1. Always check if your objects are distinct before applying standard combinatorics formulas.

Q: How do I know if there are other partitions besides (3, 3, 2)?

Great question! Since the maximum capacity is 3, the sum of three numbers must be 8. If you try to lower one number, say to 2, you would need another to be 4 to keep the sum at 8, which violates the capacity constraint. Thus, (3, 3, 2) is the only valid integer partition.

Q: Why do we multiply by 3 at the end?

Because the cars are distinct! The partition (3, 3, 2) can be assigned to the cars in three different ways: (Car1:3, Car2:3, Car3:2), (Car1:3, Car2:2, Car3:3), or (Car1:2, Car2:3, Car3:3). Each of these is a unique physical arrangement.

Q: Does the order of people inside the car matter?

In this problem, we are only concerned with which car a person is assigned to. The internal seating arrangement is not specified, so we treat the set of people in a car as a group. The multinomial coefficient handles this grouping perfectly.

Q: Is there a risk of overcounting?

Not if you define your cases clearly. By fixing the capacity distribution first and then multiplying by the number of ways to assign those capacities to the distinct cars, we ensure every unique distribution is counted exactly once.

Question

Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is

Options:

(A) 1120

(B) 3360

(C) 1680

(D) 560

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are tasked with distributing $8$ distinct individuals into $3$ distinct cars (Sedan, SUV, Hatchback). The constraint is that each car can hold a maximum of $3$ people.

Let $x_1, x_2, x_3$ represent the number of people in each car. We must satisfy the equation:

$$x_1 + x_2 + x_3 = 8$$

Subject to the constraint $x_i \le 3$ for all $i \in \{1, 2, 3\}$.

### Identifying the Partition

To satisfy the sum of $8$ with a maximum value of $3$ per variable, we examine the possible integer partitions. The only set of integers that sums to $8$ while keeping each value $\le 3$ is $\{3, 3, 2\}$.

Any other combination, such as $\{4, 2, 2\}$, violates the capacity constraint. Thus, the distribution of group sizes must be a permutation of $(3, 3, 2)$.

### The Multinomial Calculation

First, we calculate the number of ways to partition $8$ distinct people into groups of sizes $3, 3,$ and $2$. This is determined by the multinomial coefficient:

$$	ext{Ways} = \frac{8!}{3! 	imes 3! 	imes 2!}$$

Expanding the factorials, we perform the calculation:

$$	ext{Ways} = \frac{40320}{6 	imes 6 	imes 2} = \frac{40320}{72} = 560$$

This value represents the number of ways to form the groups if the car assignments were fixed for these specific sizes.

### Accounting for Distinct Cars

Because the cars are distinct, the group of $2$ people can be assigned to any of the $3$ cars. This leads to three distinct scenarios for the distribution $(x_1, x_2, x_3)$:

1. $(2, 3, 3)$
2. $(3, 2, 3)$
3. $(3, 3, 2)$

Since these scenarios are mutually exclusive, we multiply the number of ways to form the groups by the number of possible car assignments:

$$	ext{Total Ways} = 560 	imes 3 = 1680$$

### Final Result

By systematically applying the multinomial theorem and accounting for the distinct nature of the vehicles, we conclude that there are **1680** ways to organize the transport.

Animated Solution for Mathematics - Permutations and Combinations: Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is

Select Answer:

Visualized Solution

Visualizing the Setup

The Capacity Constraint

Partitioning the People

Accounting for Distinct Cars

Formula for One Arrangement

Calculating the Factorials

Evaluating One Case

Final Calculation

The Sigma Insight: Formation of Groups

Analyzing the Setup

Identifying the Partition

The Multinomial Calculation

Accounting for Distinct Cars

Final Result

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Animated Solution for Mathematics - Permutations and Combinations: Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is

Select Answer:

Visualized Solution

Visualizing the Setup

The Capacity Constraint

Partitioning the People

Accounting for Distinct Cars

Formula for One Arrangement

Calculating the Factorials

Evaluating One Case

Final Calculation

The Sigma Insight: Formation of Groups

Analyzing the Setup

Identifying the Partition

The Multinomial Calculation

Accounting for Distinct Cars

Final Result

Similar Questions

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