If three squares are selected at random from chessboard then the probability that all three squares share a common vertex is k find `[(1)/(k)]` where denotes greatest integer function

Answer

To solve the problem of finding the probability that three randomly selected squares from a chessboard share a common vertex, we will follow these steps:

### Step 1: Determine the total number of squares on the chessboard.
A standard chessboard has dimensions of 8x8, which means there are a total of:
\[ 64 \text{ squares} \quad (8 \times 8 = 64) \]

**Hint:** Remember that the total number of squares is simply the product of the number of rows and columns on the chessboard.

### Step 2: Calculate the total number of ways to choose 3 squares from the 64 squares.
The number of ways to choose 3 squares from 64 is given by the combination formula:
\[ \binom{64}{3} = \frac{64!}{3!(64-3)!} = \frac{64 \times 63 \times 62}{3 \times 2 \times 1} = 39711 \]

**Hint:** Use the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \) to find the number of ways to select squares.

### Step 3: Determine the number of ways to select 3 squares that share a common vertex.
To find the number of ways in which 3 squares can share a common vertex, we consider the arrangement of squares on the chessboard. Each vertex can be a common point for squares in a 2x2 arrangement.

- Each 2x2 block of squares has 4 vertices.
- The chessboard can accommodate \(7 \times 7 = 49\) such 2x2 blocks (since each block occupies a 2x2 area, and there are 7 positions horizontally and 7 vertically).

For each of these 2x2 blocks, we can choose any 3 squares from the 4 squares that share the common vertex. The number of ways to choose 3 squares from 4 is:
\[ \binom{4}{3} = 4 \]

Thus, the total number of ways to select 3 squares that share a common vertex is:
\[ 49 \times 4 = 196 \]

**Hint:** Consider how many 2x2 blocks can be formed on the chessboard and how many squares can be selected from each block.

### Step 4: Calculate the probability \( k \).
The probability \( k \) that all three squares share a common vertex is given by:
\[ k = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{196}{39711} \]

**Hint:** Use the formula for probability which is the ratio of favorable outcomes to total outcomes.

### Step 5: Find \( \frac{1}{k} \).
To find \( \frac{1}{k} \):
\[ \frac{1}{k} = \frac{39711}{196} \]

Calculating this gives:
\[ \frac{39711}{196} \approx 202.5 \]

### Step 6: Apply the greatest integer function.
The greatest integer function \( \lfloor x \rfloor \) gives the largest integer less than or equal to \( x \). Thus:
\[ \lfloor 202.5 \rfloor = 202 \]

### Final Answer:
The final result is:
\[ \lfloor \frac{1}{k} \rfloor = 202 \]

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