Chapter 2: Q. 2.17 (page 49)

If $8$ rooks (castles) are randomly placed on a chessboard, compute the probability that none of the rooks can capture any of the others. That is, compute the probability that no row or file contains more than one rook.

Short Answer

Expert verified

$\frac{8!}{(\begin{matrix} 64 \\ 8 \end{matrix})}$

Step by step solution

Given Information.

Let $8$ rooks (castles) be randomly placed on a chessboard,

Explanation.

We go row by row.

First, we choose a position in a row $1$ from the $8$ columns for keeping the first rooks in $(\begin{array}{l} 8 \\ 1 \end{array})$ the way.

Next, we choose a position in a row $2$ from the available $7$ columns(one column has been occupied by a rook $1$ ). This gives us $(\begin{array}{l} 7 \\ 1 \end{array})$ choices.

Continuing like this, we get $8!$ ways to put the $8$ rooks on the chessboard given the conditions.

Total ways of keeping $8$ rooks on a chessboard are simple $(\begin{matrix} 64 \\ 8 \end{matrix})$ .

Explanation.

$\frac{8!}{(\begin{matrix} 64 \\ 8 \end{matrix})}$

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If $8$ rooks (castles) are randomly placed on a chessboard, compute the probability that none of the rooks can capture any of the others. That is, compute the probability that no row or file contains more than one rook.

Short Answer

Step by step solution

Given Information.

Explanation.

Explanation.

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If 8rooks (castles) are randomly placed on a chessboard, compute the probability that none of the rooks can capture any of the others. That is, compute the probability that no row or file contains more than one rook.

Short Answer

Step by step solution

Given Information.

Explanation.

Explanation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Pure Maths

Theoretical and Mathematical Physics

Calculus

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Study anywhere. Anytime. Across all devices.

If $8$ rooks (castles) are randomly placed on a chessboard, compute the probability that none of the rooks can capture any of the others. That is, compute the probability that no row or file contains more than one rook.