Matthew works as a computer operator at a small university. One evening he finds that $12$ computer programs have been submitted earlier that day for batch processing. In how many ways can Matthew order the processing of these programs if $($a$)$ there are no restrictions? $($b$)$ he considers four of the programs higher in priority than the other eight and wants to process those four first? $($c$)$ he first separates the programs into four of top priority, five of lesser priority, and three of least priority, and he wishes to process the $12$ programs in such a way that the top$-$priority programs are processed first and the three programs of least priority are processed last?