A large apartment building is to be built using modular construction. Architects have designed three floor plans with the following specifications:

Floor Plan A: has $3$ three$-$bedrooms, $7$ two$-$bedrooms and $12$ one$-$bedroom units

Floor Plan B: has $4$ three$-$bedrooms, $4$ two$-$bedrooms and $8$ one$-$bedroom units

Floor Plan C: has $5$ three$-$bedrooms, $3$ two$-$bedrooms and $7$ one$-$bedroom units

Is it possible to design the building with exactly $66$ three$-$bedroom units, $74$ two$-$bedroom units, and $144$ one bedroom units? If so$,$ is there more than one way to do it$?$ Explain your answer.

Hint: you can use a calculator in question $3$ to perform any large scale calculation, however you are responsible for clearly identifying any variables you are using, equations and writing all details leading to the computation as well as a interpretation of result.