How many 4-digit numbers can be formed with the digits 9,8,7,5,3 when a digit may be repeated any number of times in any arrangement.

Answer

To find out how many 4-digit numbers can be formed using the digits 9, 8, 7, 5, and 3, with the allowance for repetition of digits, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Digits**: The available digits are 9, 8, 7, 5, and 3. This gives us a total of 5 different digits.

2. **Determine the Number of Positions**: We need to form a 4-digit number, which means we have 4 positions to fill.

3. **Calculate Options for Each Position**:
- For the **first position**, we can choose any of the 5 digits (9, 8, 7, 5, 3). Thus, there are 5 options.
- For the **second position**, since repetition is allowed, we can again choose any of the 5 digits. So, there are also 5 options.
- For the **third position**, similarly, we have 5 options.
- For the **fourth position**, we again have 5 options.

4. **Multiply the Options**: Since the choices for each position are independent, we multiply the number of options for each position together:
\[ \text{Total 4-digit numbers} = 5 \times 5 \times 5 \times 5 = 5^4 \]

5. **Calculate \(5^4\)**:
\[ 5^4 = 625 \]

### Final Answer:
Thus, the total number of 4-digit numbers that can be formed is **625**.

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