A logo is to be designed with two colors: an outer color and an inner color. There are a total of $170$ colors available: $59$ warm colors and $111$ cool colors. The designer must use two different colors $($no repeats$).$

How many ways can the colors be chosen...

$($a$)$ if the designer picks two colors from the entire selection of $170$ colors available?

$($b$)$ if the outer color is a warm color and the inner color is a cool color?

$($c$)$ if the designer picks a warm color and a cool color, and then assigns the two chosen colors to the two positions on the logo$?$

$($d$)$ if the outer color has to be a warm color $($and the inner color can be any of the remaining $169$ colors$)?$