From the given digits 2,2,3,3,3,4,4,4,4 how many 4-digit numbers greater than…

To be greater than 3000, the thousands digit must be 3 or 4.

• Case 1: Thousands digit = 3. Remaining digits (with counts): 2×2, 3×2, 4×4.

  Count distinct 3-digit endings from these remaining digits (no replacement):

  • Three distinct digits (2,3,4): 3! = 6.

  • One digit repeated twice and one different:

    Possible doubled digits: 2, 3, or 4. For each doubled digit, the singleton can be one of the two remaining digit types, and each such multiset yields 3 distinct permutations. So total = 3 × 2 × 3 = 18.

  • All three same: only 444 is possible here (since 4 has count ≥3), so 1.

  Total for thousands digit 3: 6 + 18 + 1 = 25.

• Case 2: Thousands digit = 4. Remaining digits (with counts): 2×2, 3×3, 4×3.

  Count distinct 3-digit endings from these remaining digits:

  • Three distinct digits (2,3,4): 3! = 6.

  • One digit repeated twice and one different:

    Possible doubled digits: 2, 3, or 4. For each doubled digit, the singleton can be one of the two remaining digit types, and each such multiset yields 3 permutations. So total = 3 × 2 × 3 = 18.

  • All three same: possible for 333 and 444 (both have count ≥3), so 2.

  Total for thousands digit 4: 6 + 18 + 2 = 26.

Adding both cases: 25 + 26 = 51.

Final answer: 51 distinct 4-digit numbers greater than 3000 can be formed.