In how many ways 48 small squasres of `1cmxx1cm` can be arranged so that the resulting area is `48cm^(2)`?

Options

A. 6
B. 4
C. 5
D. 2

Answer

To solve the problem of how many ways 48 small squares of 1 cm x 1 cm can be arranged to form an area of 48 cm², we can follow these steps:

### Step 1: Understand the problem
We need to arrange 48 squares, each with an area of 1 cm², to create a larger area of 48 cm². This means we are looking for different rectangular configurations that can be formed using these squares.

### Step 2: Determine the factors of 48
To find the different rectangular arrangements, we need to find the pairs of factors of 48. The factors of 48 are:
- 1
- 2
- 3
- 4
- 6
- 8
- 12
- 16
- 24
- 48

### Step 3: Pair the factors
Next, we will pair the factors to find the dimensions of the rectangles that can be formed:
1. (1, 48)
2. (2, 24)
3. (3, 16)
4. (4, 12)
5. (6, 8)

### Step 4: Count the unique arrangements
Each pair of factors represents a unique arrangement of squares. Since the order of dimensions in a rectangle does not matter (i.e., a rectangle of dimensions 2 x 24 is the same as 24 x 2), we only count each pair once.

From the pairs we identified, we have:
1. 1 x 48
2. 2 x 24
3. 3 x 16
4. 4 x 12
5. 6 x 8

This gives us a total of **5 unique arrangements**.

### Conclusion
Thus, the total number of ways to arrange 48 small squares of 1 cm x 1 cm to form an area of 48 cm² is **5**.

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