How many numbers between 400 and 1000 can be made with the digits 2,3, 4, 5, 6 and 0? (Repetition of digits is not allowed.)

Answer

To solve the problem of how many numbers between 400 and 1000 can be formed using the digits 2, 3, 4, 5, 6, and 0 without repetition, we can follow these steps:

### Step 1: Identify the range of numbers
We need to find three-digit numbers between 400 and 1000. This means the numbers must be in the range of 401 to 999.

### Step 2: Determine the first digit
Since the number must be at least 400, the first digit can only be 4, 5, or 6. This gives us 3 options for the first digit:
- 4
- 5
- 6

### Step 3: Choose the second digit
After choosing the first digit, we have to select the second digit. Since repetition is not allowed, we cannot use the digit chosen as the first digit.

- If the first digit is 4, the available digits for the second digit are 0, 2, 3, 5, and 6 (5 options).
- If the first digit is 5, the available digits for the second digit are 0, 2, 3, 4, and 6 (5 options).
- If the first digit is 6, the available digits for the second digit are 0, 2, 3, 4, and 5 (5 options).

In all cases, we have 5 options for the second digit.

### Step 4: Choose the third digit
After selecting the first and second digits, we will choose the third digit. Again, since repetition is not allowed, we cannot use the first or second digits.

- After choosing the first and second digits, we will have 4 remaining digits to choose from for the third digit.

### Step 5: Calculate the total combinations
Now we can calculate the total number of combinations:
- For each choice of the first digit (3 options), we have 5 options for the second digit and 4 options for the third digit.

Thus, the total number of different three-digit numbers can be calculated as follows:
\[ \text{Total Numbers} = (\text{Choices for First Digit}) \times (\text{Choices for Second Digit}) \times (\text{Choices for Third Digit}) = 3 \times 5 \times 4 \]
Calculating this gives:
\[ 3 \times 5 = 15 \]
\[ 15 \times 4 = 60 \]

### Final Answer
Therefore, the total number of numbers that can be formed between 400 and 1000 using the digits 2, 3, 4, 5, 6, and 0 without repetition is **60**.

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