How many even integers between 4000 and 7000 have four different digits?

1: Thousands digit

The number is 4-digit and must be between 4000 and 7000.
So the thousands digit can be: 4, 5, or 6.

Step 2: Units digit

The number must be even, so the units digit must be one of {0, 2, 4, 6, 8}.

3: Count cases

• Case 1: Thousands digit = 4
  Units digit ≠ 4 → possible units = {0, 2, 6, 8} → 4 choices.
  Hundreds digit = 8 choices, Tens digit = 7 choices.
  Total = 4 × 8 × 7 = 224.


• Case 2: Thousands digit = 5
  Units digit can be {0, 2, 4, 6, 8} → 5 choices.
  Hundreds digit = 8 choices, Tens digit = 7 choices.
  Total = 5 × 8 × 7 = 280.


• Case 3: Thousands digit = 6
  Units digit ≠ 6 → possible units = {0, 2, 4, 8} → 4 choices.
  Hundreds digit = 8 choices, Tens digit = 7 choices.
  Total = 4 × 8 × 7 = 224.

4: Add results

Total = 224 + 280 + 224 = 728.

Final Answer:

There are 728 such even integers.