Find the sum of all those 4 digits numbers which can be formed with the digits 1,2,3,4.

Answer

To find the sum of all four-digit numbers that can be formed using the digits 1, 2, 3, and 4, we will follow these steps:

### Step 1: Determine the total number of four-digit combinations
We can use the digits 1, 2, 3, and 4 in each of the four positions of a four-digit number. Since we can repeat digits, the total number of four-digit numbers that can be formed is calculated as:
\[ 4^4 = 256 \]
**Hint:** Remember that each digit can be chosen independently for each position.

### Step 2: Calculate how many times each digit appears in each position
Since there are 256 total combinations and we have 4 digits, each digit will appear in each position an equal number of times. To find how many times each digit appears in one position, we divide the total combinations by the number of digits:
\[ \frac{256}{4} = 64 \]
This means each digit (1, 2, 3, and 4) appears 64 times in each of the four positions.

**Hint:** Think about symmetry; since all digits are used equally, they will appear the same number of times.

### Step 3: Calculate the contribution of each digit to each position
The contribution of a digit to a specific position depends on its place value. The place values for a four-digit number are:
- Thousands place: \(10^3 = 1000\)
- Hundreds place: \(10^2 = 100\)
- Tens place: \(10^1 = 10\)
- Units place: \(10^0 = 1\)

### Step 4: Calculate the total contribution of each digit
The total contribution of each digit can be calculated as follows:
- Contribution from the thousands place: \(64 \times 1000 \times \text{(digit)}\)
- Contribution from the hundreds place: \(64 \times 100 \times \text{(digit)}\)
- Contribution from the tens place: \(64 \times 10 \times \text{(digit)}\)
- Contribution from the units place: \(64 \times 1 \times \text{(digit)}\)

Combining these, the total contribution of each digit is:
\[ 64 \times (1000 + 100 + 10 + 1) \times \text{(digit)} = 64 \times 1111 \times \text{(digit)} \]

### Step 5: Calculate the total sum for all digits
Now, we need to sum the contributions of all digits (1, 2, 3, and 4):
\[ \text{Sum} = 64 \times 1111 \times (1 + 2 + 3 + 4) \]
Calculating the sum of the digits:
\[ 1 + 2 + 3 + 4 = 10 \]
Thus, the total sum becomes:
\[ \text{Sum} = 64 \times 1111 \times 10 \]

### Step 6: Calculate the final result
Now, we compute:
\[ 64 \times 1111 \times 10 = 640 \times 1111 = 71104 \]

### Final Answer
The sum of all four-digit numbers that can be formed with the digits 1, 2, 3, and 4 is:
\[ \boxed{71104} \]

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