The number of numbers,stricily between `5000` and `10000` can be formed using the digits `1,3,5,7,9` without repetition,is

Options

A. `72`
B. `120`
C. `6`
D. `12`

Answer

To find the number of numbers strictly between 5000 and 10000 that can be formed using the digits 1, 3, 5, 7, and 9 without repetition, we can follow these steps:

### Step 1: Identify the range
The numbers we are interested in must be four-digit numbers that lie strictly between 5000 and 10000. Therefore, the first digit (thousands place) must be either 5, 7, or 9.

### Step 2: Determine possible choices for the thousands place
From the digits available (1, 3, 5, 7, 9), the valid choices for the thousands place are:
- 5
- 7
- 9

Thus, we have **3 choices** for the thousands place.

### Step 3: Choose the remaining digits
After selecting a digit for the thousands place, we have to fill the hundreds, tens, and units places. Since we cannot repeat digits, we will have 4 remaining digits to choose from (after one digit is used for the thousands place).

### Step 4: Calculate the number of combinations
For each choice of the thousands place:
- We have 4 remaining digits to choose from for the hundreds place.
- After selecting the hundreds place digit, we will have 3 digits left for the tens place.
- Finally, we will have 2 digits left for the units place.

Thus, the number of ways to fill the last three places is calculated as follows:
- Hundreds place: 4 choices
- Tens place: 3 choices
- Units place: 2 choices

The total combinations for each choice of the thousands place is:
\[ 4 \times 3 \times 2 = 24 \]

### Step 5: Total combinations
Since we have 3 choices for the thousands place, the total number of valid four-digit numbers is:
\[ 3 \times 24 = 72 \]

### Final Answer
The total number of numbers strictly between 5000 and 10000 that can be formed using the digits 1, 3, 5, 7, and 9 without repetition is **72**.

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