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

Correct Option: 72

We need to find the number of numbers of the form 0.XY, where X and Y are distinct non-zero digits. Non-zero digits from the decimal system are 1, 2, 3, 4, 5, 6, 7, 8, 9, which gives a total of 9 possible digits. Since X and Y must be distinct, we are selecting and arranging 2 digits out of 9 non-zero digits. The number of ways to choose and arrange 2 distinct digits from 9 available digits is given by the permutation formula P(n, r) = n! / (n - r)!. Here, n = 9 and r = 2. Therefore, P(9, 2) = 9 × 8 = 72. Alternatively, for the first digit X, we have 9 choices (any non-zero digit from 1 to 9). Since X and Y must be distinct, for the second digit Y, we have 8 remaining choices. Thus, the total number of such numbers is 9 × 8 = 72. Options 81, 90, and 100 are incorrect because they either include zero or allow repetition (like 0.11), which violates the condition that X and Y are distinct non-zero digits. Hence, option A is correct.