If repetition be not allowed then the number of digits lying between 5000 and 10000 which can be formed by using the digits from 1 to 9 is

Options

A. `5 times "^8 P_3`
B. `5 times "^8 C_3`
C. `5! times "^8 P_3`
D. `5! times "^8 C_3`

Answer

To solve the problem of finding the number of digits between 5000 and 10000 that can be formed using the digits from 1 to 9 without repetition, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Range**:
The numbers we are interested in are four-digit numbers that lie between 5000 and 10000. This means the first digit must be either 5, 6, 7, 8, or 9.

2. **Choose the First Digit**:
The first digit can be chosen from the set {5, 6, 7, 8, 9}. There are 5 possible choices for the first digit.

3. **Choose the Second Digit**:
After choosing the first digit, we cannot use that digit again (since repetition is not allowed). Therefore, we have 8 remaining digits to choose from for the second digit.

4. **Choose the Third Digit**:
After choosing the first and second digits, we have 7 digits left to choose from for the third digit.

5. **Choose the Fourth Digit**:
Finally, after selecting the first three digits, we have 6 digits remaining to choose from for the fourth digit.

6. **Calculate the Total Combinations**:
The total number of four-digit combinations can be calculated by multiplying the number of choices at each step:
\[ \text{Total Combinations} = (\text{Choices for 1st digit}) \times (\text{Choices for 2nd digit}) \times (\text{Choices for 3rd digit}) \times (\text{Choices for 4th digit}) \]
\[ \text{Total Combinations} = 5 \times 8 \times 7 \times 6 \]

7. **Perform the Calculation**:
\[ 5 \times 8 = 40 \]
\[ 40 \times 7 = 280 \]
\[ 280 \times 6 = 1680 \]

8. **Final Answer**:
Therefore, the total number of four-digit numbers that can be formed between 5000 and 10000 using the digits from 1 to 9 without repetition is **1680**.