Assigning Tasks Suppose that a data processing department wishes to assign five programmers to five programming tasks $($one programmer to each task$).$ Management has estimated the total number of days each programmer would take if assigned to the different jobs, and these estimates are summarized in the following table.

Task $12345$

$15025786460$

$24330705672$

Programmer $36028806668$

$45429756070$

$54532706275$

$($a$)$ Determine the assignment that minimizes the total programmer days required to complete all five jobs.

$($b$)$ How would the solution change if programmer $3$ could not be assigned to tasks $2$ or $4?$