Correct Option
The problem requires forming four-digit numbers using the digits 1, 2, 3, and 4 without repetition, while adhering to the following constraints:
- Constraint 1: Digits 2 and 3 must not appear consecutively (i.e., substrings '23' or '32' are not allowed).
- Constraint 2: Digit 1 must not be immediately followed by digit 3 (i.e., substring '13' is not allowed).
- Constraint 3: Digit 4 must not appear at the last position.
- Constraint 4: Digit 1 must not appear at the first position.
We systematically enumerate the valid permutations by considering the constraints:
Case 1: The first digit is 2. (2 _ _ _)
According to Constraint 4, the first digit cannot be 1. If the first digit is 2, then the second digit cannot be 3 (Constraint 1).
- If the number is 21_ _: The remaining digits are 3, 4.
- 2134: Violates Constraint 2 (1 is immediately followed by 3). Invalid.
- 2143: Satisfies all constraints. This is a valid number.
- If the number is 24_ _: The remaining digits are 1, 3.
- 2413: Violates Constraint 2 (1 is immediately followed by 3). Invalid.
- 2431: Satisfies all constraints. This is a valid number.
Valid numbers from Case 1: 2143, 2431 (2 numbers).
Case 2: The first digit is 3. (3 _ _ _)
If the first digit is 3, then the second digit cannot be 2 (Constraint 1).
- If the number is 31_ _: The remaining digits are 2, 4.
- 3124: Violates Constraint 3 (4 is at the last place). Invalid.
- 3142: Satisfies all constraints. This is a valid number.
- If the number is 34_ _: The remaining digits are 1, 2.
- 3412: Satisfies all constraints. This is a valid number.
- 3421: Satisfies all constraints. This is a valid number.
Valid numbers from Case 2: 3142, 3412, 3421 (3 numbers).
Case 3: The first digit is 4. (4 _ _ _)
The remaining digits are 1, 2, 3.
- If the number is 41_ _: The remaining digits are 2, 3.
- 4123: Violates Constraint 1 (2 and 3 are immediately following each other). Invalid.
- 4132: Violates Constraint 2 (1 is immediately followed by 3). Invalid.
- If the number is 42_ _: The remaining digits are 1, 3.
- 4213: Violates Constraint 2 (1 is immediately followed by 3). Invalid.
- 4231: Violates Constraint 1 (2 and 3 are immediately following each other). Invalid.
- If the number is 43_ _: The remaining digits are 1, 2.
- 4312: Satisfies all constraints. This is a valid number.
- 4321: Violates Constraint 1 (2 and 3 are immediately following each other). Invalid.
Valid numbers from Case 3: 4312 (1 number).
Combining all valid numbers from the cases:
- From Case 1: 2143, 2431
- From Case 2: 3142, 3412, 3421
- From Case 3: 4312
The total number of different numbers that can be formed is 2 + 3 + 1 = 6.
Incorrect Options
Options 2 (8), 3 (9), and 4 (None of the above) are incorrect because a systematic application of all given constraints yields exactly 6 valid numbers, as demonstrated above.