We have seen the power of polynomials and

its representation in counting, the way idea of using a polynomials coefficients

and then multiplying two polynomials leads to a different form of counting as you saw

in the chapter of generating functions, we will further look at the idea of polynomials

that originate in chessboard by using this a pawn called the rook, also called the elephant and seen how many

ways can we place a rook in chessboard and this problem has something to do with some

deep counting questions, so let us start with the introduction to the problem, look at this

chessboard this is 3 x 3 chessboard. In how many ways can I place one elephant

here, one rook here, I can place it here or here

or here and so on, there are 9 ways, one rook can be placed in

9 place, in how many ways can I place 2 rooks? Obviously you would say 9 choose 2 ways, but

note that 2 rook should be placed in such a way that when you place a rook here in the

first cell you cannot place another rook in the same row and same column, these are called

the non-taking rooks when you place 2 rooks in such a way that no 2 rooks are in the, they both are not on the same vertical line

or horizontal line, so in how many ways can I place a rook here? Once I place it in the first cell I have 4

options to place the second, so 4 ways, when I place the rook here there are 4 ways, again the second cell, I’m

going to place the rook here, there again 4 ways, as you can see the only possible ways in which

you can place 2 rooks is simply 12. In how many ways can we place 3 rooks? As you can see on this chessboard this is

one way and this is another way and there are only 2 ways, right, obliquely you can

place them 3 rooks, otherwise there is no other way, so you can

place them in 3 ways.

2 rooks – you missed (6) cases for first rook in middle row and second on bottom row (assuming bottom != top)