What is the number of numbers of the form 0·XY, where X and Y are distinct non-zero digits?
- (a) 72 ✓ UPSC's answer
- (b) 81
- (c) 90
- (d) 100
Why the answer is (a)
• The number is of the form 0.XY, where X and Y are digits in the tenths and hundredths places respectively.
• The problem states that X and Y must be non-zero digits, so the possible values for each are {1, 2, 3, 4, 5, 6, 7, 8, 9}, giving 9 options for each.
• The problem also states that X and Y must be distinct, meaning X cannot equal Y.
• For the first digit X, there are 9 possible choices (1 through 9).
• For the second digit Y, since it must be distinct from X, there are 8 remaining choices.
• The total number of such numbers is the product of the choices: 9 × 8 = 72.
Why the other options are wrong
- (b) 81
- 81 is the result of 9 × 9, which assumes X and Y can be the same, violating the distinctness condition.
- (c) 90
- 90 is the result of 10 × 9, which incorrectly includes 0 as a possible digit for X or Y, violating the non-zero condition.
- (d) 100
- 100 is the result of 10 × 10, which incorrectly allows both digits to be zero and allows them to be identical.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2022, held on 5 June 2022. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.