Correct Option (1)
The problem requires forming two-digit numbers of the form XY, where X and Y are distinct non-zero digits.
- The set of non-zero digits available is {1, 2, 3, 4, 5, 6, 7, 8, 9}. There are 9 such digits.
- For the tens digit (X), there are 9 possible choices (any of the non-zero digits).
- Since the digits X and Y must be distinct, for the units digit (Y), there are 8 remaining choices (any non-zero digit except the one chosen for X).
- The total number of such numbers is the product of the number of choices for X and Y.
- Number of ways = (Choices for X) × (Choices for Y) = 9 × 8 = 72.
- This can also be understood as the number of permutations of 9 distinct items taken 2 at a time, denoted as P(9, 2).
- P(9, 2) = 9! / (9-2)! = 9! / 7! = 9 × 8 = 72.
Incorrect Options:
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Option 2 (81): This value would be obtained if the digits X and Y were not required to be distinct. If both X and Y could be any of the 9 non-zero digits, then the total number of possibilities would be 9 × 9 = 81. However, the question explicitly states that X and Y must be distinct.
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Option 3 (90): This represents the total count of all two-digit numbers, ranging from 10 to 99. This set includes numbers with zero (e.g., 10, 20) and numbers with repeated digits (e.g., 11, 22), both of which violate the conditions of the problem (non-zero and distinct digits).
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Option 4 (100): This option is incorrect as it does not correspond to the constraints of forming two-digit numbers using distinct non-zero digits. 100 is the count of numbers from 1 to 100, or 0 to 99, neither of which aligns with the question's specific requirements.
