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

### Step 1: Count 2-digit Numbers
- 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.

**Calculation:**
- Number of choices for the first digit = 5
- Number of choices for the second digit = 4

Total 2-digit numbers = 5 × 4 = 20

### Step 2: Count 3-digit Numbers
- 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.

**Calculation:**
- Number of choices for the first digit = 5
- Number of choices for the second digit = 4
- Number of choices for the third digit = 3

Total 3-digit numbers = 5 × 4 × 3 = 60

### Step 3: Count 4-digit Numbers
- 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.

**Calculation:**
- Number of choices for the first digit = 5
- Number of choices for the second digit = 4
- Number of choices for the third digit = 3
- Number of choices for the fourth digit = 2

Total 4-digit numbers = 5 × 4 × 3 × 2 = 120

### Step 4: Total Numbers
Now, we sum up all the numbers we calculated:

Total numbers = Total 2-digit numbers + Total 3-digit numbers + Total 4-digit numbers
= 20 + 60 + 120 = 200

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

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