All possible numbers are formed using the digits `1,1,2,2,2,2,3,4,4` taken all at a time. The number of such numbers in which the odd digits occupy even places is:

Options

A. 180
B. 175
C. 160
D. 162

Answer

To solve the problem of finding the number of arrangements of the digits `1, 1, 2, 2, 2, 2, 3, 4, 4` such that the odd digits occupy the even places, we can follow these steps:

### Step 1: Identify the Digits and Their Frequencies
The digits we have are:
- 1 (occurs 2 times)
- 2 (occurs 4 times)
- 3 (occurs 1 time)
- 4 (occurs 2 times)

### Step 2: Determine the Positions
Since we are forming a 9-digit number, the positions are indexed from 1 to 9. The even positions are 2, 4, 6, and 8. This gives us a total of 4 even positions.

### Step 3: Identify the Odd Digits
The odd digits available are:
- 1 (2 times)
- 3 (1 time)

Thus, we have a total of 3 odd digits (1, 1, 3).

### Step 4: Choose Positions for Odd Digits
We need to place the 3 odd digits in the 4 even positions. The number of ways to choose 3 positions out of 4 for the odd digits can be calculated using the combination formula:
\[ \binom{4}{3} = 4 \]

### Step 5: Arrange the Odd Digits
Now, we need to arrange the 3 odd digits (1, 1, 3) in the chosen positions. The number of arrangements of these digits, accounting for the repetition of the digit 1, is given by:
\[ \frac{3!}{2!} = 3 \]

### Step 6: Fill Remaining Positions with Even Digits
After placing the odd digits, we have 6 positions left to fill with the remaining digits. The remaining digits are:
- 2 (4 times)
- 4 (2 times)

We need to arrange these 6 digits (2, 2, 2, 2, 4, 4). The number of arrangements is given by:
\[ \frac{6!}{4! \cdot 2!} \]

### Step 7: Calculate the Total Arrangements
Now, we can calculate the total number of arrangements by multiplying the number of ways to choose positions for the odd digits, the arrangements of the odd digits, and the arrangements of the even digits:
\[ \text{Total Arrangements} = \binom{4}{3} \cdot \frac{3!}{2!} \cdot \frac{6!}{4! \cdot 2!} \]

### Step 8: Perform the Calculations
Calculating each part:
1. \(\binom{4}{3} = 4\)
2. \(\frac{3!}{2!} = 3\)
3. \(\frac{6!}{4! \cdot 2!} = \frac{720}{24 \cdot 2} = 15\)

Now, multiply these results:
\[ \text{Total Arrangements} = 4 \cdot 3 \cdot 15 = 180 \]

Thus, the final answer is:
\[ \boxed{180} \]