The sum of all the 4-digit distibct numbers that can be formed with the digit 1,2,2and 3 is

Answer

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

### Step 1: Count the total distinct 4-digit numbers
We need to calculate the total number of distinct arrangements of the digits 1, 2, 2, and 3. Since the digit '2' is repeated, we use the formula for permutations of multiset:

\[ \text{Total arrangements} = \frac{n!}{n_1! \cdot n_2! \cdot \ldots} \]

Where \( n \) is the total number of items to arrange, and \( n_1, n_2, \ldots \) are the counts of each distinct item. Here, we have:

- Total digits = 4 (1, 2, 2, 3)
- Counts: 1 appears once, 2 appears twice, and 3 appears once.

Thus, the total arrangements are:

\[ \text{Total arrangements} = \frac{4!}{1! \cdot 2! \cdot 1!} = \frac{24}{2} = 12 \]