Answer to Question #256191 in Algebra for Sohaila

Question

The EML Train Company makes five different-sized train cars: a 1-person car, a 2-person car, a 3-person car, a 4-person car, and a 5-person car. These cars can be connected to form trains that hold different numbers of people. The cars have special hitches so that two cars can only be connected if their numbers are consecutive numbers. For example, you can connect a 2-passenger and 3-passenger car, but not a 2-passenger and 4-passenger car.

Part 1

Try to build some trains with the special hitches. You can use only these five types of cars to build trains, and you can use at most one of each type of car in each train. Each train must consist of at least two cars.

• What are all the different numbers of people that the EML Train Company can build trains to hold? Explain how you found this.
• What are the different numbers of people that the EML Train Company can not build trains to hold? Explain how you found this.

Answer 1

Combination of 2 cars chosen from 5 cars

$_5C_2= \frac{5!}{2!3!}=10$

Combination of 3 cars chosen from 5 cars

$_5C_3=\frac{5!}{3!2!}=10$

Combination of 4 cars chosen from 5 cars

$= _5C_4= \frac{5!}{4!1!}=5$

Combination of 5 cars chosen from 5 cars =$_5C_5=\frac{5!}{5!0!}=1$

Part A

TRAINS THAT CAN BE BUILD

For 2 cars, combinations of number of people are;

1 and 2, 2 and 3, 3 and 4, 4 and 5

For 3 cars, combination of numbers of people are;

(1,2 and 3), (2,3 and 4), (3,4 and 5)

For 4 cars, combination of numbers of people are;

(1,2,3 and 4), (2,3,4 and 5)

For 5 cars, combination of numbers of people are;

(1,2,3,4 and 5)

All the different number of people that the company can build trains to hold are;

3,5,6,7,9,10,12,14 and 15

Part B

TRAINS THAT CANNOT BE BUILD

For 2 cars

1 and 3, 1 and 4, 1 and 5, 2 and 4, 2 and 5, 3 and 5

For 3 cars

(1,2 and 4), (1,2 and 5), (1,3 and 4), (1,3 and 5), (1,4 and 5), (2,3 and 5), (2,4 and 5)

For 4 cars

(1,2,3 and 5), (1,2,4 and 5), (1,3,4 and 5)

The number of different people that the company cannot build trains to hold are;

(4,8,11and 13)