Four-digit numbers are to be formed using the digits 1, 2, 3 and 4; and none of these four digits are repeated in any manner. Further, 1. 2 and 3 are not to immediately follow each other 2. 1 is not to be immediately followed by 3 3. 4 is not to appear at the last place 4. 1 is not to appear at the first place How many different numbers can be formed ?
- (a) 6 ✓ UPSC's answer
- (b) 8
- (c) 9
- (d) None of the above
Why the answer is (a)
• Rules: 2 and 3 never adjacent; 1 never immediately followed by 3; 4 not last; 1 not first.
• Check the permutations starting with 2, 3 or 4 against these rules.
• Starting with 2: 2143 and 2431 qualify (2413 fails, since 1 is followed by 3).
• Starting with 3: 3142, 3412 and 3421 qualify (3124 ends in 4; others put 2 next to 3).
• Starting with 4: only 4312 qualifies (4123 and 4321 put 2 next to 3; 4213 and 4132 put 3 after 1; 4231 has 2 next to 3).
• Total = 2 + 3 + 1 = 6.
• Hence 6, option (a).
Why the other options are wrong
- (b) 8
- Option 8 is incorrect because only 6 valid permutations satisfy all four constraints.
- (c) 9
- Option 9 is incorrect because it likely includes invalid numbers like 4123 or 4213 which violate the adjacency rules.
- (d) None of the above
- Option 'None of the above' is incorrect because 6 is a valid count and is listed in option (a).
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.