Step-by-step Solution:

1. Thousand Place Digits: The thousand place digits can be 4, 5, or 6. 2. Case 1: Thousand Place Digit is Even (4 or 6): - Thousand Place: 2 choices (4 or 6). - Hundreds Place: 8 choices (0-9, excluding the thousand digit). - Tens Place: 7 choices (0-9, excluding the thousand and hundreds digits). - Units Place: 4 choices (0, 2, 4, 6, 8, excluding the thousand, hundreds, and tens digits). Total for this case: \[ 2 \times 8 \times 7 \times 4 = 448 \] 3. Case 2: Thousand Place Digit is Odd (5): - Thousand Place: 1 choice (5). - Hundreds Place: 8 choices (0-9, excluding the thousand digit). - Tens Place: 7 choices (0-9, excluding the thousand and hundreds digits). - Units Place: 5 choices (0, 2, 4, 6, 8, excluding the thousand, hundreds, and tens digits). Total for this case: \[ 1 \times 8 \times 7 \times 5 = 280 \] 4. Total Number of Even Integers: Add the totals from both cases: \[ 448 + 280 = 728 \] Therefore, the total number of even integers between 4000 and 7000 with four different digits is: \[ \boxed{728} \]