Sixteen squares of dimension 1 x 1 are chosen at random on a chessboard, having 64 squares of ision 1 x 1. What is the probability that a square of 4 x 4 is formed?

To find the probability of selecting 16 squares on a chessboard such that they form a \(4 \times 4\) square, let's break down the problem step by step:

### Step 1: Total number of ways to choose 16 squares
The chessboard has 64 squares (since it's an \(8 \times 8\) chessboard). The total number of ways to randomly choose 16 squares from these 64 squares is given by the combination:
\[
\text{Total ways} = \binom{64}{16}
\]
This represents the number of ways to choose any 16 squares out of 64 without any restrictions.

### Step 2: Number of favorable outcomes (i.e., forming a \(4 \times 4\) square)
To form a \(4 \times 4\) square, we need to select exactly the 16 squares that make up such a square. Let's count how many distinct \(4 \times 4\) squares can be formed on an \(8 \times 8\) chessboard.

- Along the rows, you can place the top row of the \(4 \times 4\) square starting from row 1 to row 5 (because if you start in row 6 or higher, the square will go off the board).
- Along the columns, you can place the leftmost column of the \(4 \times 4\) square starting from column 1 to column 5.

Thus, the number of distinct \(4 \times 4\) squares that can be formed on the chessboard is:
\[
\text{Number of favorable squares} = 5 \times 5 = 25
\]

### Step 3: Probability calculation
The probability of selecting 16 squares such that they form a \(4 \times 4\) square is the ratio of favorable outcomes to total outcomes. Therefore, the probability is:
\[
\text{Probability} = \frac{\text{Number of favorable squares}}{\text{Total ways}} = \frac{25}{\binom{64}{16}}
\]

This expression gives the exact probability, but calculating \(\binom{64}{16}\) will require computational tools because it's a large number.

Would you like me to compute this value for you, or would you prefer further clarification on the steps?

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### Related Questions:
1. What is the probability of choosing 16 squares randomly such that they form a \(3 \times 3\) square?
2. How many total ways are there to arrange 2x2 squares on an \(8 \times 8\) chessboard?
3. What is the number of ways to form a \(4 \times 4\) square on a \(10 \times 10\) grid?
4. How does the probability change if you were to randomly select 9 squares on a chessboard?
5. How do combination formulas help in calculating probabilities in grid problems?

**Tip:** In probability problems involving grids, visualizing smaller examples (like a \(3 \times 3\) grid) can help you understand patterns before scaling up.

Sixteen squares of dimension 1 x 1 are chosen at random on a chessboard, having 64 squares of dimension 1 x 1. What is the probability that a square of 4 x 4 is formed?

Learn how to calculate the probability of randomly choosing 16 squares on an 8x8 chessboard to form a 4x4 square. This problem involves combinatorics and probability concepts, perfect for Grades 10-12 students.

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