Relationship between X and Y:

• X is a two-digit number. Y is formed by interchanging the digits of X.
• X  = 10a + b, (where a is the tens digit and b is the units digit), then Y = 10b + a.

Calculating X + Y:

• X + Y = (10a + b) + (10b + a)
• X + Y = 11a + 11b
• X + Y = 11(a + b)

Now, since (X + Y) is the greatest two-digit number, it must be 99.

• So, we have: 11(a + b) = 99
• Means, a + b = 9

Now, we need to find the possible values of a and b that satisfy this equation while keeping them within the two-digit range (0 to 9).

• The possible combinations for a and b are: (1, 8), (2, 7), (3, 6), (4, 5), (5, 4), (6, 3), (7, 2), (8, 1)
• So, the number of possible values of X is 8.
• Therefore, the correct answer is: (d)