There are three work centers $($A$,$ B$,$ and C$)$ behind the financial aid counter at a nearby university. They can each fit into any of three office spaces $(1,2,$ and $3)$ off the corridor behind the desk. There is no student contact in these areas, only workers. The distance $1-2$ is $20$ feet, $2-3$ is $30$ feet, and $1-3$ is $50$ feet. The matrix of work $($trips per day$)$ at the three centers are shown in the following table. Remember that each trip must be a round$-$trip $($from $1$ to $2$ and back, for example$).$

$($a$)$ How many possible assignments are there? List them.

$($b$)$ Calculate the total distance traveled in each of these assignments.

$($c$)$ Which assignment minimizes distance traveled?

A

B

C

A

$--$

$45$

$60$

B

$20$

$--$

$0$

C

$0$

$25$

$--$