Solution:

To find the required probability, we need to count the number of ways in which two different numbers can be selected from the set {0, 1, 2, 3, ......., 10) such that their sum as well as absolute difference are both multiple of 4.

Let's first count the number of ways in which two different numbers can be selected from the set {0, 1, 2, 3, ......., 10) such that their sum is a multiple of 4.

- Selecting two numbers whose sum is a multiple of 4:

There are 3 numbers in the given set that are multiples of 4: 0, 4, and 8. If we select any two of these numbers, their sum will be a multiple of 4. Thus, there are 3C2 ways to select two numbers whose sum is a multiple of 4.

- Selecting two numbers whose sum is not a multiple of 4:

There are 8 numbers in the given set that are not multiples of 4: 1, 2, 3, 5, 6, 7, 9, and 10. If we select any two of these numbers, their sum will not be a multiple of 4. Thus, there are 8C2 ways to select two numbers whose sum is not a multiple of 4.

Thus, the total number of ways to select two different numbers from the set {0, 1, 2, 3, ......., 10) such that their sum is a multiple of 4 is:

3C2 + 8C2 = 3 + 28 = 31

Now, let's count the number of ways in which two different numbers can be selected from the set {0, 1, 2, 3, ......., 10) such that their absolute difference is a multiple of 4.

- Selecting two numbers whose absolute difference is a multiple of 4:

There are 3 numbers in the given set that are multiples of 4: 0, 4, and 8. If we select any two of these numbers, their absolute difference will be a multiple of 4. Thus, there are 3C2 ways to select two numbers whose absolute difference is a multiple of 4.

- Selecting two numbers whose absolute difference is not a multiple of 4:

There are 7 pairs of numbers in the given set whose absolute difference is not a multiple of 4: (0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3), and (9, 10). If we select any two numbers from the remaining 4 numbers (5, 6, 7, and 9), their absolute difference will not be a multiple of 4. Thus, there are 4C2 ways to select two numbers whose absolute difference is not a multiple of 4.

Thus, the total number of ways to select two different numbers from the set {0, 1, 2, 3, ......., 10) such that their absolute difference is a multiple of 4 is:

3C2 + 4C2 = 3 + 6 = 9

Finally, the required probability is the ratio of the number of