Math Problem Statement

Recall that a rook in chess attacks along rows and columns. Define R_n to be the number of ways of arranging n rooks on an n x n chessboard so that no two can attack each other and their placement is symmetric about the diagonal from lower left to upper right. Which is a correct formula for R_n?

Solution

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Problem:

Recall that a rook in chess attacks along rows and columns. Define RnR_n to be the number of ways of arranging nn rooks on an n×nn \times n chessboard so that no two can attack each other, and their placement is symmetric about the diagonal from the lower left to the upper right. Which is a correct formula for RnR_n?

The options provided are:

  1. Rn=n!2R_n = \frac{n!}{2}
  2. Rn=n(n1)2R_n = \frac{n(n-1)}{2}
  3. Rn=Rn1+(n1)Rn2R_n = R_{n-1} + (n-1) \cdot R_{n-2}
  4. Rn=nRn1+(n1)Rn2R_n = n \cdot R_{n-1} + (n-1) \cdot R_{n-2}

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Math Problem Analysis

Mathematical Concepts

Combinatorics
Symmetry in arrangements
Chessboard problems

Formulas

R_n = R_{n-1} + (n-1) * R_{n-2}
R_n = n * R_{n-1} + (n-1) * R_{n-2}

Theorems

Principle of Permutations and Combinations

Suitable Grade Level

College level or advanced high school