How many numbers are there between 3000 and 4000 which can be formed with the digits 3,4,5,6,7,8 and repetition of digits is not allowed?

Answer

To solve the problem of how many numbers can be formed between 3000 and 4000 using the digits 3, 4, 5, 6, 7, and 8 without repetition, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Range**:
We need to form 4-digit numbers between 3000 and 4000. Therefore, the first digit (thousands place) must be 3.

**Hint**: The first digit determines the range of the number.

2. **Fix the First Digit**:
Since the first digit must be 3, we have:
- Thousands place: 3

3. **Choose the Remaining Digits**:
After fixing the first digit as 3, we can use the remaining digits: 4, 5, 6, 7, and 8. This gives us 5 options for the second digit (hundreds place).

**Hint**: Count the remaining digits after fixing the first digit.

4. **Select the Second Digit**:
We can choose any of the 5 remaining digits (4, 5, 6, 7, 8) for the hundreds place. So, we have 5 choices.

5. **Select the Third Digit**:
After choosing the second digit, we will have 4 digits left to choose from for the tens place.

**Hint**: The number of choices decreases as you select more digits.

6. **Select the Fourth Digit**:
After selecting the third digit, we will have 3 digits left for the units place.

7. **Calculate the Total Combinations**:
The total number of combinations can be calculated by multiplying the number of choices for each digit:
- Total combinations = 1 (fixed first digit) × 5 (choices for second digit) × 4 (choices for third digit) × 3 (choices for fourth digit)

\[ \text{Total combinations} = 1 \times 5 \times 4 \times 3 = 60 \]

### Final Answer:
There are **60 numbers** that can be formed between 3000 and 4000 using the digits 3, 4, 5, 6, 7, and 8 without repetition.