How many 4-digit numbers are there with no digit repeated?

Answer

To find how many 4-digit numbers can be formed with no repeated digits, we can follow these steps:

### Step 1: Determine the first digit
The first digit of a 4-digit number cannot be 0 (as it would then be a 3-digit number). Therefore, we can choose the first digit from the numbers 1 to 9. This gives us 9 options for the first digit.

**Hint:** Remember that the first digit cannot be 0 in a 4-digit number.

### Step 2: Determine the second digit
After choosing the first digit, we have already used one digit. We can now choose the second digit from the remaining digits, which includes 0 and the other digits except the first digit. This gives us 9 options for the second digit.

**Hint:** After selecting the first digit, consider all remaining digits including 0 for the second position.

### Step 3: Determine the third digit
For the third digit, we have already used two digits. Therefore, we can choose the third digit from the remaining 8 digits. This gives us 8 options for the third digit.

**Hint:** Keep track of how many digits have been used so far to know how many options are left.

### Step 4: Determine the fourth digit
Finally, for the fourth digit, we have used three digits already. Thus, we can choose the fourth digit from the remaining 7 digits. This gives us 7 options for the fourth digit.

**Hint:** Again, remember to count how many digits have been used to find the remaining options.

### Step 5: 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
- Choices for the fourth digit: 7

So, the total number of 4-digit numbers with no repeated digits is:
\[ 9 \times 9 \times 8 \times 7 \]

Calculating this gives:
\[ 9 \times 9 = 81 \\ 81 \times 8 = 648 \\ 648 \times 7 = 4536 \]

Thus, the total number of 4-digit numbers with no digit repeated is **4536**.

**Final Answer:** 4536