Сколькими способами можно расположить на шахматной доске две ладьи так, чтобы одна не била другую. Сколькими способами можно расположить на шахматной доске две ладьи так, чтобы одна не била другую.
Сколькими способами можно расположить на шахматной доске две ладьи так, чтобы одна не била другую. Сколькими способами можно расположить на шахматной доске две ладьи так, чтобы одна не била другую.
Сколькими способами можно расположить на шахматной доске две ладьи так, чтобы одна не била другую. Сколькими способами можно расположить на шахматной доске две ладьи так, чтобы одна не била другую. ...
GPT-4.1
To solve this problem, we'll determine the number of ways to place two rooks on a standard 8×8 chessboard such that they do not attack each other. In chess, a rook attacks all squares along its row and column. Therefore, two rooks do not attack each other if they are placed on different rows and different columns.
**Step 1: Calculate the Total Number of Ways to Place Two Rooks**
First, calculate the total number of ways to place two rooks on the chessboard without any restrictions.
- There are 64 squares on the chessboard.
- The number of ways to choose any two distinct squares is given by the combination:
\[
\text{Total ways} = \binom{64}{2} = \frac{64 \times 63}{2} = 2016
\]
**Step 2: Calculate the Number of Attacking Positions**
Next, calculate the number of ways the two rooks **do attack** each other (i.e., are in the same row or the same column).
- **Rooks in the Same Row:**
- There are 8 rows on the chessboard.
- In each row, there are 8 squares.
- The number of ways to choose two squares in the same row is:
\[
\binom{8}{2} = 28
\]
- Total ways where rooks are in the same row:
\[
8 \text{ rows} \times 28 = 224 \text{ ways}
\]
- **Rooks in the Same Column:**
- Similarly, there are 8 columns.
- Total ways where rooks are in the same column:
\[
8 \text{ columns} \times 28 = 224 \text{ ways}
\]
- **Adjust for Overcounting:**
- Positions where rooks are both in the same row **and** the same column (i.e., the same square) have been counted in both the row and column counts.
- However, since we are choosing two distinct squares, there's no possibility of the rooks occupying the same square.
- Therefore, the total number of attacking positions is:
\[
\text{Attacking ways} = 224 (\text{same row}) + 224 (\text{same column}) = 448
\]
**Step 3: Calculate the Number of Non-Attacking Positions**
Subtract the number of attacking positions from the total number of positions:
\[
\text{Non-attacking ways} = \text{Total ways} - \text{Attacking ways} = 2016 - 448 = 1568
\]
**Answer:**
There are **1568** ways to arrange two rooks on a chessboard so that one does not attack the other.
To solve this problem, we'll determine the number of ways to place two rooks on a standard 8×8 chessboard such that they do not attack each other. In chess, a rook attacks all squares along its row and column. Therefore, two rooks do not attack each other if they are placed on different rows and different columns.
**Step 1: Calculate the Total Number of Ways to Place Two Rooks**
First, calculate the total number of ways to place two rooks on the chessboard without any restrictions.
- There are 64 squares on the chessboard.
- The number of ways to choose any two distinct squares is given by the combination:
\[
\text{Total ways} = \binom{64}{2} = \frac{64 \times 63}{2} = 2016
\]
**Step 2: Calculate the Number of Attacking Positions**
Next, calculate the number of ways the two rooks **do attack** each other (i.e., are in the same row or the same column).
- **Rooks in the Same Row:**
- There are 8 rows on the chessboard.
- In each row, there are 8 squares.
- The number of ways to choose two squares in the same row is:
\[
\binom{8}{2} = 28
\]
- Total ways where rooks are in the same row:
\[
8 \text{ rows} \times 28 = 224 \text{ ways}
\]
- **Rooks in the Same Column:**
- Similarly, there are 8 columns.
- Total ways where rooks are in the same column:
\[
8 \text{ columns} \times 28 = 224 \text{ ways}
\]
- **Adjust for Overcounting:**
- Positions where rooks are both in the same row **and** the same column (i.e., the same square) have been counted in both the row and column counts.
- However, since we are choosing two distinct squares, there's no possibility of the rooks occupying the same square.
- Therefore, the total number of attacking positions is:
\[
\text{Attacking ways} = 224 (\text{same row}) + 224 (\text{same column}) = 448
\]
**Step 3: Calculate the Number of Non-Attacking Positions**
Subtract the number of attacking positions from the total number of positions:
\[
\text{Non-attacking ways} = \text{Total ways} - \text{Attacking ways} = 2016 - 448 = 1568
\]
**Answer:**
There are **1568** ways to arrange two rooks on a chessboard so that one does not attack the other.
To solve this problem, we'll determine the number of ways to place two rooks on a standard 8×8 chessboard such that they do not attack each other. In chess, a rook attacks all squares along its row and column. Therefore, two rooks do not attack each other if they are placed on different rows and different columns.
Step 1: Calculate the Total Number of Ways to Place Two Rooks
First, calculate the total number of ways to place two rooks on the chessboard without any restrictions.
There are 64 squares on the chessboard.
The number of ways to choose any two distinct squares is given by the combination:
