Why study only rook polynomials? In introductory combinatorics, there is an emphasis on rook polynomials. But what is the significance of only considering rook polynomials? Why not consider "knight polynomials" or "bishop polynomials?" 
 A: This is a very honest and short answer.
Going through rook polynomials teaches good combinatorial skills while solving otherwise very challenging problems. I would say they form a smaller part of a potentially much larger focus on generating functions. But the problem with generating functions is that half the time, they feel either contrived or products of divine inspiration. I consider generating functions very useful (I have used them both for pleasure and research), but largely out of place in many introductory courses.
So let's say we admit that such polynomials are good. Why rooks? Others are too hard. Well, pawns are boring. Bishop polynomials are the same thing as interlaced rook polynomials (one for the light squares and one for the dark squares). Knight polynomials would be nontrivial, I think. A quick look on google tells me that there are many sorts of boards where knight polynomials are known, but the general case is too tedious. Whether or not this is true, I would say that it is certainly true that knights are hard. Continuing on, queens and kings are also both simple.
In short, they are comparatively easy and useful in incorporating fundamental skills.
