To solve the problem of forming 8-digit numbers using the digits 1, 1, 2, 2, 2, 3, 4, 4, where the odd digits do not occupy odd positions, we can follow these steps:
### Step-by-Step Solution:
1. **Identify the Digits**:
The digits available are 1, 1, 2, 2, 2, 3, 4, 4.
- Odd digits: 1, 1, 3 (total 3 odd digits)
- Even digits: 2, 2, 2, 4, 4 (total 5 even digits)
2. **Determine Positions**:
In an 8-digit number, the positions are numbered from 1 to 8. The odd positions are 1, 3, 5, and 7, while the even positions are 2, 4, 6, and 8.
3. **Odd Digits Cannot Occupy Odd Positions**:
Since the odd digits (1, 1, 3) cannot occupy the odd positions (1, 3, 5, 7), they must occupy the even positions (2, 4, 6, 8).
4. **Choose Positions for Odd Digits**:
We have 4 even positions (2, 4, 6, 8) and we need to place 3 odd digits in these positions. We can choose any 3 out of the 4 even positions to place the odd digits.
- The number of ways to choose 3 positions from 4 is given by \( \binom{4}{3} = 4 \).
5. **Arrange the Odd Digits**:
The arrangement of the odd digits (1, 1, 3) in the chosen positions can be calculated using the formula for permutations of multiset:
\[
\text{Arrangements} = \frac{3!}{2!} = 3
\]
(since the digit '1' is repeated twice).
6. **Fill Remaining Positions with Even Digits**:
After placing the odd digits, we have 1 even position left, and we need to fill this with even digits (2, 2, 2, 4, 4).
The remaining even digits will be 2, 2, 4, 4 (since we used one of the even digits to fill the last position).
- The number of arrangements of these even digits is:
\[
\frac{4!}{3! \cdot 1!} = 4
\]
7. **Combine the Results**:
Now, we multiply the number of ways to choose the positions for the odd digits, the arrangements of the odd digits, and the arrangements of the even digits:
\[
\text{Total Numbers} = \text{Ways to choose positions} \times \text{Arrangements of odd digits} \times \text{Arrangements of even digits}
\]
\[
= 4 \times 3 \times 4 = 48
\]
### Final Answer:
The total number of 8-digit numbers formed using the digits 1, 1, 2, 2, 2, 3, 4, 4, where the odd digits do not occupy odd positions is **48**.
