find the sum of all the four digit numbers which are formed by the digits 1,2,5,6

Options

A. a)933510
B. b)93324
C. c)65120
D. d)8400

Answer

To find the sum of all four-digit numbers that can be formed using the digits 1, 2, 5, and 6, we can follow these steps:

### Step 1: Determine the number of permutations
Since we have 4 distinct digits (1, 2, 5, and 6), the total number of four-digit numbers that can be formed is given by the factorial of the number of digits:
\[ \text{Number of permutations} = 4! = 24 \]

### Step 2: Calculate the sum of the digits
Next, we need to find the sum of the digits we are using:
\[ \text{Sum of digits} = 1 + 2 + 5 + 6 = 14 \]

### Step 3: Contribution of each digit in each place
Each digit will appear in each place (thousands, hundreds, tens, and units) an equal number of times. Since there are 24 total permutations and 4 places, each digit will appear in each place:
\[ \text{Number of times each digit appears in each place} = \frac{24}{4} = 6 \]

### Step 4: Calculate the contribution of each place
Now, we calculate the contribution of each digit in each place:
- **Thousands place contribution**: Each digit contributes \(1000 \times 6\) times its value.
- **Hundreds place contribution**: Each digit contributes \(100 \times 6\) times its value.
- **Tens place contribution**: Each digit contributes \(10 \times 6\) times its value.
- **Units place contribution**: Each digit contributes \(1 \times 6\) times its value.

The total contribution from all places can be calculated as follows:
\[ \text{Total contribution} = 6 \times (1000 + 100 + 10 + 1) \times \text{Sum of digits} \]
\[ = 6 \times 1111 \times 14 \]

### Step 5: Calculate the final sum
Now we compute:
\[ 6 \times 1111 = 6666 \]
\[ 6666 \times 14 = 93324 \]

Thus, the sum of all four-digit numbers formed by the digits 1, 2, 5, and 6 is:
\[ \boxed{93324} \]