The sum of all five digit numbers formed with the digits 1,2,3,4,5 without repetition of digits are

Options

A. 3,00,000
B. 3,60,000
C. 3,90,000
D. None of these

Answer

To find the sum of all five-digit numbers formed with the digits 1, 2, 3, 4, and 5 without repetition, we can follow these steps:

### Step 1: Calculate the total number of five-digit numbers
The total number of five-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5 without repetition is given by the factorial of the number of digits, which is 5! (5 factorial).

\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

**Hint:** Remember that n! (n factorial) represents the number of ways to arrange n distinct objects.

### Step 2: Determine how many times each digit appears in each place value
Since we have 5 digits and we are forming 120 different numbers, each digit will appear in each place (units, tens, hundreds, thousands, ten-thousands) equally.

The number of times each digit appears in each place value can be calculated by dividing the total number of arrangements by the number of digits:

\[ \text{Occurrences per digit} = \frac{120}{5} = 24 \]

**Hint:** Each digit appears the same number of times in each position due to symmetry in arrangements.

### Step 3: Calculate the contribution of each digit to the total sum
Each digit contributes to the total sum based on its position. The place values for a five-digit number are:

- Ten-thousands place: \(10,000\)
- Thousands place: \(1,000\)
- Hundreds place: \(100\)
- Tens place: \(10\)
- Units place: \(1\)

The total contribution of each digit can be calculated as follows:

\[ \text{Total contribution of a digit} = \text{Occurrences} \times (\text{Value of place}) \times \text{Digit} \]

Calculating the total contribution for all digits:

\[ \text{Total contribution} = 24 \times (10000 + 1000 + 100 + 10 + 1) \times (1 + 2 + 3 + 4 + 5) \]

Calculating the sum of the place values:

\[ 10000 + 1000 + 100 + 10 + 1 = 11111 \]

Calculating the sum of the digits:

\[ 1 + 2 + 3 + 4 + 5 = 15 \]

Thus, the total contribution becomes:

\[ \text{Total contribution} = 24 \times 11111 \times 15 \]

**Hint:** Break down the calculations step by step to avoid mistakes.

### Step 4: Calculate the final sum
Now we can calculate the final sum:

\[ \text{Total contribution} = 24 \times 11111 \times 15 \]

Calculating \(11111 \times 15\):

\[ 11111 \times 15 = 166665 \]

Now multiplying by 24:

\[ 24 \times 166665 = 3999960 \]

### Final Answer
The sum of all five-digit numbers formed with the digits 1, 2, 3, 4, and 5 without repetition is:

\[ \boxed{3999960} \]