While writing all the numbers from 700 to 1000, how many numbers occur in which the digit at hundred's place is greater than the digit at ten's place, and the digit at ten's place is greater than the digit at unit's place?
- (a) 61
- (b) 64
- (c) 85 ✓ UPSC's answer
- (d) 91
Why the answer is (c)
• The hundreds digit H is 7, 8 or 9 (1000 does not qualify), and we need H > T > U for the tens digit T and units digit U.
• Any 2 different digits smaller than H give exactly one valid (T, U) pair, so the count is the number of ways of choosing 2 digits from those below H.
• H = 7: choose 2 of 0–6 = 21; H = 8: choose 2 of 0–7 = 28; H = 9: choose 2 of 0–8 = 36.
• Total = 21 + 28 + 36 = 85, option (c).
Why the other options are wrong
- (a) 61
- 61 undercounts: the three cases give 21, 28 and 36.
- (b) 64
- 64 undercounts: the three cases give 21, 28 and 36.
- (d) 91
- 91 overcounts: the three cases give 21, 28 and 36.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2018, held on 3 June 2018. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.