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

Explanation

To find how many 3-digit numbers can be formed with no digit repeated, we can follow these steps:

### Step 1: Determine the first digit
The first digit of a 3-digit number cannot be 0 (as it would then be a 2-digit number). Therefore, we can choose the first digit from the digits 1 to 9. This gives us 9 options for the first digit.

**Hint:** Remember that the first digit of a 3-digit number cannot be 0.

### Step 2: Determine the second digit
Once the first digit is chosen, we need to choose the second digit. The second digit can be any digit from 0 to 9, except for the digit already chosen as the first digit. This gives us 9 options for the second digit.

**Hint:** The second digit can include 0, but it cannot be the same as the first digit.

### Step 3: Determine the third digit
For the third digit, we can choose any digit from 0 to 9, excluding the two digits that have already been chosen. This gives us 8 options 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 calculate the total number of 3-digit numbers with no repeated digits by multiplying the number of choices for each digit:

\[ \text{Total 3-digit numbers} = (\text{Choices for first digit}) \times (\text{Choices for second digit}) \times (\text{Choices for third digit}) \]
\[ \text{Total 3-digit numbers} = 9 \times 9 \times 8 \]

### Step 5: Perform the multiplication
Calculating this gives:

\[ 9 \times 9 = 81 \]
\[ 81 \times 8 = 648 \]

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

**Final Answer:** There are 648 different 3-digit numbers with no digit repeated.