The sum of 5 digit numbers in which only odd digits occur without any repetition is

Options

A. 277775
B. 555550
C. 1111100
D. None of these

Answer

To find the sum of all 5-digit numbers formed using only odd digits (1, 3, 5, 7, 9) without any repetition, we can follow these steps:

### Step 1: Identify the odd digits
The odd digits available are: 1, 3, 5, 7, 9.

### Step 2: Calculate the total number of 5-digit combinations
Since we are using all 5 digits and they must be unique, the total number of 5-digit combinations can be calculated using the factorial of the number of digits:
\[ 5! = 120 \]
This means there are 120 different 5-digit numbers that can be formed.

### Step 3: Contribution of each digit to each position
Each digit will appear in each of the 5 positions (ten-thousands, thousands, hundreds, tens, and units) equally. Since there are 120 total numbers and 5 positions, each digit will appear in each position:
\[ \frac{120}{5} = 24 \text{ times} \]

### Step 4: Calculate the sum of the odd digits
Now, we need to find the sum of the odd digits:
\[ 1 + 3 + 5 + 7 + 9 = 25 \]

### Step 5: Calculate the contribution to each position
Each digit contributes to the total based on its position:
- For the ten-thousands place, the contribution is \(24 \times 25 \times 10,000\)
- For the thousands place, the contribution is \(24 \times 25 \times 1,000\)
- For the hundreds place, the contribution is \(24 \times 25 \times 100\)
- For the tens place, the contribution is \(24 \times 25 \times 10\)
- For the units place, the contribution is \(24 \times 25 \times 1\)

### Step 6: Calculate the total sum
Now, we can calculate the total sum:
\[ \text{Total Sum} = 24 \times 25 \times (10,000 + 1,000 + 100 + 10 + 1) \]
Calculating the sum inside the parentheses:
\[ 10,000 + 1,000 + 100 + 10 + 1 = 11,111 \]
Now substituting back:
\[ \text{Total Sum} = 24 \times 25 \times 11,111 \]

### Step 7: Final Calculation
Calculating \(24 \times 25\):
\[ 24 \times 25 = 600 \]
Now, multiplying by 11,111:
\[ 600 \times 11,111 = 6,666,600 \]

Thus, the sum of all 5-digit numbers formed using only odd digits without repetition is:
\[ \boxed{6,666,600} \]