Four-digit numbers are to be formed using the digits 1, 2, 3 and 4; and none of these four digits are repeated

Question

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?

Accepted Answer

Constraints: (i) 1 cannot be first; (ii) 4 cannot be last; (iii) the digrams "12", "23", and "13" are forbidden.

List the permutations of {1,2,3,4} that start with 2, 3, or 4 (since 1 cannot be first) and check each:

Starting with 2:
• 2143 — valid (no forbidden digram; 4 not last).
• 2431 — valid.
• 2134 (contains 13), 2314 (23), 2341 (23), 2413 (13) — all invalid.

Starting with 3:
• 3142 — valid.
• 3241 — valid.
• 3421 — valid.
• 3124 (12), 3214 (4 last), 3412 (12) — invalid.

Starting with 4:
• 4321 — valid.
• 4123 (12), 4132 (13), 4213 (13), 4231 (23), 4312 (12) — invalid.

Valid numbers: 2143, 2431, 3142, 3241, 3421, 4321 — a total of 6.

Therefore, the answer is A.

Answer

6

Answer

8

Answer

9

Answer

None of the above

Explanation

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