<p>How many even numbers of four digits can be made with the digits 0, 3, 5, 4? Find the sum of the numbers.</p>
Step-by-Step Solution
Key Concept: Even four-digit numbers require the last digit to be 0 or 4 (the only even digits available). Since we can't repeat digits, we must count arrangements with 0 at the end separately from those with 4 at the end, because a 0 cannot be in the first position.
<p><strong>Step 1: Identify constraints for even four-digit numbers</strong></p><p>Last digit must be 0 or 4 (only even digits). First digit cannot be 0. All digits must be distinct (no repetition).</p><p><strong>Step 2: Case 1 - Last digit is 0</strong></p><p>Format: _ _ _ 0. Remaining digits {3, 5, 4} fill first three positions.</p><p>Number of arrangements = 3! = 6</p><p>Numbers: 3540, 3450, 4530, 4350, 5340, 5430</p><p><strong>Step 3: Case 2 - Last digit is 4</strong></p><p>Format: _ _ _ 4. Remaining digits {0, 3, 5} fill first three positions, but first digit ≠ 0.</p><p>First position: 2 choices (3 or 5)</p><p>Second position: 2 choices (remaining two digits)</p><p>Third position: 1 choice</p><p>Number of arrangements = 2 × 2 × 1 = 4</p><p>Numbers: 3054, 3504, 5034, 5304</p><p><strong>Step 4: Total count</strong></p><p>Total even numbers = 6 + 4 = <strong>10</strong></p><p><strong>Step 5: Calculate sum</strong></p><p><strong>Case 1 (ending in 0):</strong> Sum = 3540 + 3450 + 4530 + 4350 + 5340 + 5430 = 26640</p><p><strong>Case 2 (ending in 4):</strong> Sum = 3054 + 3504 + 5034 + 5304 = 16896</p><p><strong>Total sum = 26640 + 16896 = <strong>43536</strong></strong></p>
Correct Answer: 10 even numbers; sum = 43536