How many numbers between 10 and 10,000 can be formed by using the digits 1, 2, 3, 4, 5 if
Digits can be repeated

Answer

To find how many numbers between 10 and 10,000 can be formed using the digits 1, 2, 3, 4, and 5 with repetition allowed, we can break it down into three cases based on the number of digits in the numbers: 2-digit numbers, 3-digit numbers, and 4-digit numbers.

### Step 1: Count 2-digit numbers
- A 2-digit number can be represented as \( AB \), where \( A \) is the first digit (tens place) and \( B \) is the second digit (units place).
- The first digit \( A \) can be any of the 5 digits (1, 2, 3, 4, 5).
- The second digit \( B \) can also be any of the 5 digits (1, 2, 3, 4, 5).
- Therefore, the total number of 2-digit numbers is:
\[ 5 \times 5 = 25 \]

### Step 2: Count 3-digit numbers
- A 3-digit number can be represented as \( ABC \), where \( A \) is the first digit (hundreds place), \( B \) is the second digit (tens place), and \( C \) is the third digit (units place).
- The first digit \( A \) can be any of the 5 digits.
- The second digit \( B \) can also be any of the 5 digits.
- The third digit \( C \) can also be any of the 5 digits.
- Therefore, the total number of 3-digit numbers is:
\[ 5 \times 5 \times 5 = 125 \]

### Step 3: Count 4-digit numbers
- A 4-digit number can be represented as \( ABCD \), where \( A \) is the first digit (thousands place), \( B \) is the second digit (hundreds place), \( C \) is the third digit (tens place), and \( D \) is the fourth digit (units place).
- The first digit \( A \) can be any of the 5 digits.
- The second digit \( B \) can also be any of the 5 digits.
- The third digit \( C \) can also be any of the 5 digits.
- The fourth digit \( D \) can also be any of the 5 digits.
- Therefore, the total number of 4-digit numbers is:
\[ 5 \times 5 \times 5 \times 5 = 625 \]

### Step 4: Total numbers between 10 and 10,000
- Now, we add the total numbers from each case:
\[ \text{Total} = \text{2-digit numbers} + \text{3-digit numbers} + \text{4-digit numbers} \]
\[ \text{Total} = 25 + 125 + 625 = 775 \]

Thus, the total number of numbers that can be formed between 10 and 10,000 using the digits 1, 2, 3, 4, and 5 with repetition allowed is **775**.