How many numbers between 5000 and 10,000 can be formed using the digits 1,2,3,4,5,6,7,8,9, each digit appearing not more than once in each number?

Options

A. `5xx""^(8)P_(3)`
B. `5xx""^(8)C_(8)`
C. `5!xx""^(8)P_(3)`
D. `5!xx""^(8)C_(3)`

Answer

To solve the problem of how many numbers between 5000 and 10,000 can be formed using the digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 (with each digit appearing not more than once), we can follow these steps:

### Step 1: Determine the range of numbers
We need to find four-digit numbers that lie between 5000 and 10,000. Therefore, the first digit must be 5, 6, 7, 8, or 9.

### Step 2: Choose the first digit
The first digit can be any of the following: 5, 6, 7, 8, or 9. This gives us **5 options** for the first digit.

### Step 3: Choose the second digit
After choosing the first digit, we cannot use that digit again. Since we initially have 9 digits (1 to 9), after using one digit for the first position, we are left with 8 remaining digits. Therefore, we have **8 options** for the second digit.

### Step 4: Choose the third digit
Similarly, after selecting the first and second digits, we have used 2 digits. This leaves us with 7 digits available to choose from for the third position. Hence, we have **7 options** for the third digit.

### Step 5: Choose the fourth digit
Finally, after selecting the first three digits, we have used 3 digits, leaving us with 6 remaining digits. Thus, we have **6 options** for the fourth digit.

### Step 6: Calculate the total combinations
Now, we can calculate the total number of four-digit numbers that can be formed:
\[ \text{Total Numbers} = (\text{Choices for 1st digit}) \times (\text{Choices for 2nd digit}) \times (\text{Choices for 3rd digit}) \times (\text{Choices for 4th digit}) \]
\[ \text{Total Numbers} = 5 \times 8 \times 7 \times 6 \]

### Step 7: Perform the multiplication
Calculating the above expression:
\[ 5 \times 8 = 40 \]
\[ 40 \times 7 = 280 \]
\[ 280 \times 6 = 1680 \]

Thus, the total number of four-digit numbers that can be formed between 5000 and 10,000 using the digits 1 to 9, with no repetitions, is **1680**.

### Final Answer
The total number of numbers between 5000 and 10,000 that can be formed is **1680**.

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