car race problem

I think I muffed the numbers for configurations if team cars are indistinguishable from each other as it should be exactly 1/32 of the answer to the actual problem stated..

Anyways, I probably wasn't terribly clear on what I meant by 'chains'. The concept applied here is abstract in nature.

The starting premise is that each car is beside a car of the other team. We thus have 5 'pairs' of cars in the line-up. In examining how many configurations there are, one can start by examining how many ways you can pair the cars before placing them, without regard to the question of which row and right/left just yet.

Thus far, nothing here to trip anyone up. It's all relatively simple to understand. However, it gets a little abstract here as I try to count the potential pairing combinations.

So what I do now is to start with Car 1-1. I ask who he is paired with, or which car is beside his. Call this Car B.

I then ask myself who Car B's teammate is paired with, calling it Car C, and continue with this line of questions until inevitably someone gets paired with Car 1-2. This entire group of cars is considered one 'chain'.

It may be that I have accounted for all 10 cars, or it may be that there is another 'chain' of cars.

Anyways, if you've followed me to that point, you can perhaps understand what I mean by them now. But what analysis can be made of all this? How do we use this, exactly?

Well, each pair of cars is one link in a chain. And since no car is paired with their teammate, each chain has at least 2 links.

So with 5 links, we either have one 'chain' with 5 links, or 2 'chains' with 3 and 2 links respectively, and we can get numbers of pairings that fall under each and add them together.

I hope this helps walk you through my logic on that part.

And I have figured out where I went wrong on my answer for configurations if you cannot distinguish team cars from each other.

For chains of 5, there are 12 (of half) the pairings.

For chains of 3+2, selecting the pair automatically selects how the trio is paired off (divide pairings by half).

For chains of 3+2, the 2-chain links cannot be distinguished, which means there are only half as many ways to place the cars for each combination of pairings.

(The easier way to think about it is that for each configuration for this question, there are 32 ways to place the cars if you can distinguish teammates.)