We always try and tread lightly when talking about assignments to avoid the appearance of impropriety with regards to the integrity of the exam. Plus, I'm not familiar with Factory Homes, so I'm not sure what Product and CompletedDesign are? But is the problem with the many-to-many scenario? Certainly there is nothing unusual about a many-to-many relationship. Do you have something that you could compare it to so we can generate a bit more discussion?
A person can design a designer car.He can use various components and can make various version of product(car).
When he is done he can finalize a design by choosing one of the designed car out of so many that he has designed.
How can this relationship can be represented?
I think it should be
Finalcardesign  ------[1..*] desginedcars
Any thoughts about probable scenario if the representation is as following
This scenario can occur if i make many final designs but i may not have purchased any of them yet.
Once i decide that i am ready to purchase one of the final design the state of that design will be "Purchased".
Unless your use case doesn't specify that your designedcar SHOULD be assigned to a specific finalcardesign. If you build your car, store it in some kind of cache so that you can assign it later to more than one finalcardesign, your scenario it's ok.