The sum of all the four digit even numbers which can be formed by using the digits 0, 1, 2, 3, 4 and 5 if repetition of digits is allowed is

Options

A. 1765980
B. 1756980
C. 1769580
D. 1759680

Answer

To find the sum of all the four-digit even numbers that can be formed using the digits 0, 1, 2, 3, 4, and 5 with repetition allowed, we can follow these steps:

### Step 1: Identify the possible digits
The digits available are 0, 1, 2, 3, 4, and 5. Since we are forming four-digit numbers, the first digit cannot be 0.

### Step 2: Determine the last digit
Since we are looking for even numbers, the last digit must be one of the even digits available: 0, 2, or 4.

### Step 3: Calculate the total number of four-digit even numbers
1. **Case 1: Last digit is 0**
- First digit options: 1, 2, 3, 4, 5 (5 options)
- Second digit options: 0, 1, 2, 3, 4, 5 (6 options)
- Third digit options: 0, 1, 2, 3, 4, 5 (6 options)
- Total combinations = 5 * 6 * 6 = 180

2. **Case 2: Last digit is 2**
- First digit options: 1, 2, 3, 4, 5 (5 options)
- Second digit options: 0, 1, 2, 3, 4, 5 (6 options)
- Third digit options: 0, 1, 2, 3, 4, 5 (6 options)
- Total combinations = 5 * 6 * 6 = 180

3. **Case 3: Last digit is 4**
- First digit options: 1, 2, 3, 4, 5 (5 options)
- Second digit options: 0, 1, 2, 3, 4, 5 (6 options)
- Third digit options: 0, 1, 2, 3, 4, 5 (6 options)
- Total combinations = 5 * 6 * 6 = 180

### Step 4: Calculate the contribution of each digit place
1. **Sum of the last digit contributions**
- Last digit = 0 contributes: 0 * 180 = 0
- Last digit = 2 contributes: 2 * 180 = 360
- Last digit = 4 contributes: 4 * 180 = 720
- Total contribution from last digits = 0 + 360 + 720 = 1080

2. **Sum of the hundreds place contributions**
- Each digit (0 to 5) appears equally in the hundreds place across all combinations.
- Sum of digits = 0 + 1 + 2 + 3 + 4 + 5 = 15
- Each digit appears in 180 combinations (60 for each last digit case).
- Contribution from hundreds place = 15 * 30 * 100 = 45000

3. **Sum of the tens place contributions**
- Similar to the hundreds place, the contribution is the same.
- Contribution from tens place = 15 * 30 * 10 = 4500

4. **Sum of the thousands place contributions**
- Only digits 1 to 5 can be in the thousands place.
- Sum of digits = 1 + 2 + 3 + 4 + 5 = 15
- Contribution from thousands place = 15 * 36 * 1000 = 540000

### Step 5: Calculate the total sum
Total sum = Contribution from thousands + hundreds + tens + units
= 540000 + 45000 + 4500 + 1080 = 589580

### Final Answer
The sum of all the four-digit even numbers that can be formed is **589580**.

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