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Chapter 8: Problem 3

a) Find the rook polynomial for the standard \(8 \times 8\) chessboard. b) Answer part (a) with 8 replaced by \(n\), for \(n \in \mathbf{Z}^{+}\).

Short Answer

Expert verified
The rook polynomial for an \(8 \times 8\) chessboard is given by \[R_8(x) = \sum_{r=0}^{8} {{8}\choose{r}} r! x^{r}\]. The rook polynomial for an \(n \times n\) chessboard, where \(n \in \mathbf{Z}^{+}\), is given by \[R_n(x) = \sum_{r=0}^{n} {{n}\choose{r}} r! x^{r}\].

Step by step solution

01

Calculate the Rook Polynomial for 8x8 Chessboard

An \(8 \times 8\) chessboard has 8 rows and 8 columns. A rook can either be placed or not on any square. If rooks are placed on \(r\) squares, then there are \({{8}\choose{r}}\) ways to choose these squares and \(r!\) ways to place the rooks. Summing this over \(r\) from 0 to 8, we get the rook polynomial \[R_8(x) = \sum_{r=0}^{8} {{8}\choose{r}} r! x^{r}\].
02

Generalize the Result

If we replace 8 by \(n\), the rook polynomial for an \(n \times n\) chessboard is given by \[R_n(x) = \sum_{r=0}^{n} {{n}\choose{r}} r! x^{r}\]. This formula can be used for any \(n \in \mathbf{Z}^{+}\).

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