How many numbers between 10 and 10,000 can be formed by using the digits 1, 2, 3, 4, 5 if
No digit is repeated in any number

Answer

To solve the problem of how many numbers between 10 and 10,000 can be formed using the digits 1, 2, 3, 4, and 5 without repeating any digit, we will break it down into cases based on the number of digits in the numbers we can form.

### Step 1: Count the Two-Digit Numbers
For two-digit numbers, we have two places to fill:
- The first digit (tens place) can be any of the 5 digits (1, 2, 3, 4, 5).
- The second digit (units place) can be any of the remaining 4 digits.

Thus, the total number of two-digit numbers is calculated as follows:
\[ \text{Total two-digit numbers} = 5 \times 4 = 20 \]

### Step 2: Count the Three-Digit Numbers
For three-digit numbers, we have three places to fill:
- The first digit (hundreds place) can be any of the 5 digits.
- The second digit (tens place) can be any of the remaining 4 digits.
- The third digit (units place) can be any of the remaining 3 digits.

Thus, the total number of three-digit numbers is calculated as follows:
\[ \text{Total three-digit numbers} = 5 \times 4 \times 3 = 60 \]

### Step 3: Count the Four-Digit Numbers
For four-digit numbers, we have four places to fill:
- The first digit (thousands place) can be any of the 5 digits.
- The second digit (hundreds place) can be any of the remaining 4 digits.
- The third digit (tens place) can be any of the remaining 3 digits.
- The fourth digit (units place) can be any of the remaining 2 digits.

Thus, the total number of four-digit numbers is calculated as follows:
\[ \text{Total four-digit numbers} = 5 \times 4 \times 3 \times 2 = 120 \]

### Step 4: Consider Five-Digit Numbers
Since we are looking for numbers between 10 and 10,000, we cannot form any five-digit numbers using the digits 1, 2, 3, 4, and 5, as the smallest five-digit number (12345) exceeds 10,000.

### Step 5: Calculate the Total
Now, we will sum the total numbers from each case:
\[ \text{Total numbers} = \text{Total two-digit numbers} + \text{Total three-digit numbers} + \text{Total four-digit numbers} \]
\[ \text{Total numbers} = 20 + 60 + 120 = 200 \]

Thus, the total number of numbers that can be formed between 10 and 10,000 using the digits 1, 2, 3, 4, and 5 without repeating any digit is **200**.