How many numbers lying between `999` and `10000` can be formed with the help of the digit `0,2,3,6,7,8` when the digits are not to be repeated

Options

A. 100
B. 200
C. 300
D. 400

Answer

To solve the problem of how many numbers can be formed between 999 and 10,000 using the digits 0, 2, 3, 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 that lie between 999 and 10,000. Therefore, all the numbers we form will be 4-digit numbers.

2. **Determine the First Digit**:
- The first digit of a 4-digit number cannot be 0 (as it would not be a 4-digit number).
- The possible digits for the first position are 2, 3, 6, 7, and 8. This gives us **5 options** for the first digit.

3. **Determine the Second Digit**:
- After choosing the first digit, we can use 0 for the second digit. However, we cannot repeat the digit used in the first position.
- Since we have already used one digit, we have **5 remaining digits** (including 0) to choose from for the second digit.

4. **Determine the Third Digit**:
- For the third digit, we cannot use the digits already chosen for the first and second positions.
- This leaves us with **4 remaining digits** to choose from for the third digit.

5. **Determine the Fourth Digit**:
- For the fourth digit, we again cannot use the digits already chosen for the first, second, and third positions.
- This leaves us with **3 remaining digits** to choose from for the fourth digit.

6. **Calculate the Total Combinations**:
- The total number of combinations can be calculated by multiplying the number of choices for each digit:
\[ \text{Total Combinations} = (\text{Choices for 1st digit}) \times (\text{Choices for 2nd digit}) \times (\text{Choices for 3rd digit}) \times (\text{Choices for 4th digit}) \]
\[ = 5 \times 5 \times 4 \times 3 \]
\[ = 300 \]

### Final Answer:
Thus, the total number of 4-digit numbers that can be formed between 999 and 10,000 using the digits 0, 2, 3, 6, 7, and 8 without repetition is **300**.

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