To solve the problem of how many nine-digit numbers can be formed using the digits 2, 2, 3, 3, 5, 5, 8, 8, 8, with the condition that odd digits occupy even positions, we can follow these steps:
### Step 1: Identify the digits and their frequencies
We have the following digits:
- Odd digits: 3, 3, 5, 5 (total of 4 odd digits)
- Even digits: 2, 2, 8, 8, 8 (total of 5 even digits)
### Step 2: Determine the positions
In a nine-digit number, the positions are numbered from 1 to 9. The even positions are 2, 4, 6, and 8. Thus, we have 4 even positions available for the odd digits.
### Step 3: Arrange the odd digits in the even positions
We need to arrange the 4 odd digits (3, 3, 5, 5) in the 4 even positions. The number of ways to arrange these digits can be calculated using the formula for permutations of multiset:
\[
\text{Number of arrangements} = \frac{n!}{n_1! \cdot n_2!}
\]
where \(n\) is the total number of items to arrange, and \(n_1, n_2\) are the frequencies of the repeated items.
Here, we have:
- Total odd digits \(n = 4\) (two 3's and two 5's)
- Frequencies: \(n_1 = 2\) (for 3's), \(n_2 = 2\) (for 5's)
So, the number of arrangements is:
\[
\text{Number of arrangements} = \frac{4!}{2! \cdot 2!} = \frac{24}{4} = 6
\]
### Step 4: Arrange the even digits in the remaining positions
Now, we have 5 even digits (2, 2, 8, 8, 8) to fill the remaining 5 odd positions (1, 3, 5, 7, 9). The number of ways to arrange these digits is:
\[
\text{Number of arrangements} = \frac{5!}{2! \cdot 3!}
\]
where:
- Total even digits \(n = 5\) (two 2's and three 8's)
- Frequencies: \(n_1 = 2\) (for 2's), \(n_2 = 3\) (for 8's)
Calculating this gives:
\[
\text{Number of arrangements} = \frac{120}{2 \cdot 6} = \frac{120}{12} = 10
\]
### Step 5: Calculate the total arrangements
Finally, to find the total number of nine-digit numbers, we multiply the number of arrangements of odd digits by the number of arrangements of even digits:
\[
\text{Total arrangements} = 6 \times 10 = 60
\]
Thus, the total number of nine-digit numbers that can be formed under the given conditions is **60**.
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