$($A$)$ A manufacturing plant produces three product lines in one of its plants: A$,B,$ and $C.$ Each product line has multiple models: $5$ models within product line A$,3$ models within B$,$ and $4$ within $C.$ Total annual production quantity of all models of product $A$ is $2,000$ units, $1,500$ units for product $B,$ and $800$ units for product $C.$ Determine the number of $($a$)$ different product models and $($b$)$ total quantity of products made annually in this plant.