How many numbers are there between 100 and 300 which either begin with or end with 2 ?
- (a) 110 ✓ UPSC's answer
- (b) 111
- (c) 112
- (d) None of the above
Why the answer is (a)
• The range of numbers is from 101 to 299 (inclusive), as 'between 100 and 300' typically excludes the endpoints in such counting problems, or we can count 100-300 and adjust if necessary, but standard interpretation for 'between' in UPSC CSAT often implies exclusive or we check the specific digits. Let's assume the range is 101 to 299. Actually, let's look at the numbers 100 to 300. The question asks for numbers *between* 100 and 300. This usually means 101 to 299. Let's verify the count for 101-299.
• Numbers beginning with 2: These are 200 to 299. There are 100 such numbers (200, 201, ..., 299).
• Numbers ending with 2: These are 102, 112, 122, ..., 192, 202, 212, ..., 292. In the range 101-299, the numbers ending in 2 are 102, 112, 122, 132, 142, 152, 162, 172, 182, 192 (10 numbers in the 100s) and 202, 212, 222, 232, 242, 252, 262, 272, 282, 292 (10 numbers in the 200s). Total ending with 2 is 20.
• We must subtract the overlap (numbers that both begin with 2 AND end with 2) to avoid double counting. The numbers that begin with 2 and end with 2 are 202, 212, 222, 232, 242, 252, 262, 272, 282, 292. There are 10 such numbers.
• Using the Principle of Inclusion-Exclusion: Count = (Numbers beginning with 2) + (Numbers ending with 2) - (Numbers beginning with 2 AND ending with 2).
• Calculation: 100 + 20 - 10 = 110.
• Therefore, there are 110 such numbers.
Why the other options are wrong
- (b) 111
- Option (b) 111 is incorrect because it likely results from an error in counting the overlap or the total set, such as including 100 or 300 incorrectly or miscounting the intersection.
- (c) 112
- Option (c) 112 is incorrect because it overcounts the valid numbers, possibly by failing to subtract the intersection of numbers starting and ending with 2 correctly.
- (d) None of the above
- Option (d) is incorrect because the calculated count of 110 matches option (a), so 'None of the above' is not applicable.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2016, held on 7 August 2016. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.