Solution:

To find out the number of different differences that can be obtained by taking only 2 numbers at a time from 3, 5, 2, 10 and 15, we need to follow the steps below:

Step 1: Find all possible pairs of numbers that can be formed using the given numbers.

The possible pairs of numbers are:

- 3 and 5
- 3 and 2
- 3 and 10
- 3 and 15
- 5 and 2
- 5 and 10
- 5 and 15
- 2 and 10
- 2 and 15
- 10 and 15

Step 2: Find the difference between each pair of numbers.

The differences between the pairs of numbers are:

- 5 - 3 = 2
- 2 - 3 = -1
- 10 - 3 = 7
- 15 - 3 = 12
- 2 - 5 = -3
- 10 - 5 = 5
- 15 - 5 = 10
- 10 - 2 = 8
- 15 - 2 = 13
- 15 - 10 = 5

Step 3: Count the number of different differences.

There are 6 positive differences: 2, 7, 12, 5, 8, and 13.

There are 6 negative differences: -1, -3, -5, -8, -10, and -13.

So, the total number of different differences is:

6 positive differences + 6 negative differences = 12 different differences

However, we need to consider the absolute value of the differences, as we are only interested in the number of different differences, regardless of whether they are positive or negative.

So, the total number of different differences is:

12 different differences x 2 (to consider the absolute value) = 24 different differences

Finally, we need to add the difference of 0, which can be obtained by taking any number with itself.

So, the total number of different differences that can be obtained by taking only 2 numbers at a time from 3, 5, 2, 10 and 15 is:

24 different differences + 1 (the difference of 0) = 25 different differences

Answer: Option C (1440)

720