A transporter truck has three compact cars,

[Guest]

hey I have another question:

A transporter truck has three compact cars, a station wagon, and a minivan on its trailer. In how many ways can the driver load the shipment so that one of the heavier vehicles is directly over the rear axle of the trailer?

thanks in advance.

 

[soroban]

Hello, Anna!

A transporter truck has three compact cars, a station wagon, and a minivan on its trailer.
In how many ways can the driver load the shipment so that one of the heavier vehicles
is directly over the rear axle of the trailer?

I assume that heavier vehicles are the station wagon \(\displaystyle (S)\) and the minivan \(\displaystyle (M)\)
. .and the compact cars \(\displaystyle (c)\) are the lightest.
I further assume that the three compacts cars are identical (interchangeable).


There are only two cases to consider:

[1] \(\displaystyle S\) is on the rear axle.
Then the other vehicles \(\displaystyle \{c,c,c,M\}\) can be arranged in \(\displaystyle 4\) ways.
(There are 4 choices for placing the \(\displaystyle M\); the \(\displaystyle c\)'s go in the remaining places.)

[2] \(\displaystyle M\) is on the rear axle.
Then the other vehicles \(\displaystyle \{c,c,c,S\}\) can be arranged in \(\displaystyle 4\) ways.

Therefore, there are: \(\displaystyle \,4\,+\,4\:=\:\fbox{8\text{ ways}}\)