The sum of all 4 digit number that can be formed by using the digits 2 ,4, 6 ,8 .

Options

A. 133320
B. 533280
C. 53328
D. none of these

Answer

To find the sum of all 4-digit numbers that can be formed using the digits 2, 4, 6, and 8, we can follow these steps:

### Step 1: Determine the number of permutations
Since we have 4 distinct digits (2, 4, 6, and 8), the total number of 4-digit numbers that can be formed is given by the number of permutations of these 4 digits. This can be calculated using the formula for permutations:

\[ n! = 4! = 4 \times 3 \times 2 \times 1 = 24 \]

### Step 2: Contribution of each digit in each place value
Each digit will appear in each of the four positions (thousands, hundreds, tens, and units) an equal number of times. Since there are 4 digits and 24 total arrangements, each digit will appear in each position:

\[ \text{Number of times each digit appears in each position} = \frac{24}{4} = 6 \]

### Step 3: Calculate the contribution of each digit
Now, we will calculate the contribution of each digit to the total sum based on its position:

- **Thousands place contribution**: Each digit contributes \(1000 \times 6\)
- **Hundreds place contribution**: Each digit contributes \(100 \times 6\)
- **Tens place contribution**: Each digit contributes \(10 \times 6\)
- **Units place contribution**: Each digit contributes \(1 \times 6\)

### Step 4: Sum of the digits
Now, we need to find the sum of the digits:

\[ 2 + 4 + 6 + 8 = 20 \]

### Step 5: Calculate total contribution from all digits
Now, we can calculate the total contribution from all digits for each place value:

- **Total contribution from the thousands place**:
\[ 20 \times 1000 \times 6 = 120000 \]

- **Total contribution from the hundreds place**:
\[ 20 \times 100 \times 6 = 12000 \]

- **Total contribution from the tens place**:
\[ 20 \times 10 \times 6 = 1200 \]

- **Total contribution from the units place**:
\[ 20 \times 1 \times 6 = 120 \]

### Step 6: Sum all contributions
Finally, we sum all the contributions from each place value:

\[ 120000 + 12000 + 1200 + 120 = 133320 \]

Thus, the sum of all 4-digit numbers that can be formed using the digits 2, 4, 6, and 8 is **133320**.

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