How many 3-digit numbers are there with no digit repeated?

Explanation

To find the total number of 3-digit numbers with no repeated digits, we can follow these steps:

### Step 1: Determine the choices for the first digit
The first digit of a 3-digit number cannot be zero (as it would then be a 2-digit number). Therefore, we have the digits 1 through 9 available for the first digit. This gives us 9 possible choices.

**Hint:** Remember that the first digit cannot be zero in a 3-digit number.

### Step 2: Determine the choices for the second digit
Once the first digit is chosen, we have used one digit. This means we can now use any of the remaining digits, including zero, but excluding the digit already used. Since we had 10 digits (0-9) and have used one, we have 9 digits left to choose from for the second digit.

**Hint:** Consider that the second digit can include zero, but must be different from the first digit.

### Step 3: Determine the choices for the third digit
After selecting the first and second digits, we have used two digits. Therefore, we have 8 remaining digits to choose from for the third digit.

**Hint:** The third digit must be different from both the first and second digits.

### Step 4: Calculate the total number of combinations
Now we can multiply the number of choices for each digit together:
- Choices for the first digit: 9
- Choices for the second digit: 9
- Choices for the third digit: 8

So, the total number of 3-digit numbers with no repeated digits is:
\[ 9 \times 9 \times 8 = 648 \]

### Final Answer
Thus, the total number of 3-digit numbers with no digit repeated is **648**.

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