What is the number of five -digit numbers formed with 0,1,2,3 ,4 without any repetition of digits?

Options

A. 24
B. 48
C. 96
D. 120

Answer

To find the number of five-digit numbers that can be formed using the digits 0, 1, 2, 3, and 4 without any repetition of digits, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Digits**: The available digits are 0, 1, 2, 3, and 4.

2. **Determine the First Digit**:
- The first digit of a five-digit number cannot be 0 (as it would then be a four-digit number).
- Therefore, the possible choices for the first digit are 1, 2, 3, or 4. This gives us **4 options**.

3. **Determine the Remaining Digits**:
- After choosing the first digit, we have 4 digits left (including 0).
- We can choose any of the remaining 4 digits for the second position. This gives us **4 options** for the second digit.
- After choosing the second digit, we will have 3 digits left for the third position. This gives us **3 options** for the third digit.
- For the fourth position, we will have 2 digits left, giving us **2 options**.
- Finally, for the fifth position, we will have 1 digit left, giving us **1 option**.

4. **Calculate the Total Combinations**:
- The total number of five-digit numbers can be calculated by multiplying the number of options for each digit:
\[ \text{Total Numbers} = (\text{Choices for 1st digit}) \times (\text{Choices for 2nd digit}) \times (\text{Choices for 3rd digit}) \times (\text{Choices for 4th digit}) \times (\text{Choices for 5th digit}) \]
\[ \text{Total Numbers} = 4 \times 4 \times 3 \times 2 \times 1 \]

5. **Perform the Multiplication**:
- Calculating this gives:
\[ 4 \times 4 = 16 \]
\[ 16 \times 3 = 48 \]
\[ 48 \times 2 = 96 \]
\[ 96 \times 1 = 96 \]

6. **Final Result**:
- Therefore, the total number of five-digit numbers that can be formed is **96**.

### Final Answer:
The number of five-digit numbers formed with the digits 0, 1, 2, 3, and 4 without any repetition of digits is **96**.

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