What is the number of numbers of the form 0. X Y, where X and Y are distinct non-zero digits?

The correct option is 72.

Let us break down the solution step-by-step to understand why this is the correct answer.

We are asked to find the number of numbers of the form 0.XY, where X and Y represent the digits in the tenths and hundredths decimal places, respectively.

The question specifies two main conditions for the digits X and Y:
1. X and Y must be non-zero digits.
2. X and Y must be distinct (different from each other).

Let us analyze the choices for each digit:

Step 1: Determine the choices for the digit X
The possible single-digit numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
Since X must be a non-zero digit, X cannot be 0.
Therefore, the possible choices for X are {1, 2, 3, 4, 5, 6, 7, 8, 9}.
This gives us a total of 9 possible options for the digit X.

Step 2: Determine the choices for the digit Y
Similarly, Y must also be a non-zero digit, so Y cannot be 0.
Additionally, the digits X and Y must be distinct (X ≠ Y). This means whichever digit is chosen for X cannot be chosen for Y.
Since one non-zero digit has already been assigned to X, we must exclude it from the 9 available non-zero digits.
Therefore, the number of choices remaining for Y is:
$9-1=8$ choices.

Step 3: Calculate the total number of combinations
Using the fundamental counting principle, the total number of ways to form the number 0.XY is the product of the number of choices for X and the number of choices for Y:
Total numbers = (Choices for X) × (Choices for Y)
$\text{Total numbers}=9\times 8=72$

Thus, there are exactly 72 numbers of the form 0.XY satisfying the given conditions.