Correct Option

Let X be a two-digit number. It can be represented as 10a + b, where 'a' is the tens digit and 'b' is the units digit. For X to be a two-digit number, 'a' must be a non-zero digit (i.e., a ∈ {1, 2, ..., 9}).

Let Y be another two-digit number formed by interchanging the digits of X. Thus, Y can be represented as 10b + a. For Y to also be a two-digit number, 'b' must be a non-zero digit (i.e., b ∈ {1, 2, ..., 9}).

The problem states that (X + Y) is the greatest two-digit number. The greatest two-digit number is 99.

Therefore, we can set up the equation:

X + Y = (10a + b) + (10b + a)

X + Y = 11a + 11b

X + Y = 11(a + b)

Given that X + Y = 99:

11(a + b) = 99

a + b = 9

We need to find pairs of digits (a, b) such that their sum is 9, and both 'a' and 'b' are non-zero (as established for X and Y to be two-digit numbers). The possible pairs (a, b) are:

If a = 1, then b = 8. This gives X = 18. (Y = 81)
If a = 2, then b = 7. This gives X = 27. (Y = 72)
If a = 3, then b = 6. This gives X = 36. (Y = 63)
If a = 4, then b = 5. This gives X = 45. (Y = 54)
If a = 5, then b = 4. This gives X = 54. (Y = 45)
If a = 6, then b = 3. This gives X = 63. (Y = 36)
If a = 7, then b = 2. This gives X = 72. (Y = 27)
If a = 8, then b = 1. This gives X = 81. (Y = 18)

If a = 9, then b would be 0. In this case, X = 90, but Y would be 09 = 9, which is not a two-digit number. Therefore, (9, 0) is not a valid pair for (a, b). Similarly, (0, 9) is not valid as X would be 09 = 9.

Counting the valid possibilities, there are 8 possible values for X.

Incorrect Options

Options 1 (2), 2 (4), and 3 (6) are incorrect because the calculation demonstrates that there are 8 distinct two-digit numbers X that satisfy all the given conditions, including the requirement that both X and Y are two-digit numbers.