# Placing rooks on the chessboard

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.