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.