How many different nine digit numbers can be formed with the number 223355888 by rearranging its digits sothat the odd digit occupy even positions ?

Options

A. 16
B. 36
C. 60
D. 180

Answer

To solve the problem of how many different nine-digit numbers can be formed with the digits 223355888, ensuring that the odd digits occupy even positions, we can follow these steps:

### Step 1: Identify the digits and their frequencies
We have the digits:
- 2 (occurs 2 times)
- 3 (occurs 2 times)
- 5 (occurs 2 times)
- 8 (occurs 3 times)

### Step 2: Determine the positions for odd and even digits
In a nine-digit number, the positions are numbered from 1 to 9. The even positions are 2, 4, 6, and 8 (4 positions), and the odd positions are 1, 3, 5, 7, and 9 (5 positions).

Since we need to place the odd digits (3 and 5) in the even positions, we will place them in the 4 even positions available.

### Step 3: Arrange the odd digits in the even positions
The odd digits we have are 3 and 5. We have:
- 3 (occurs 2 times)
- 5 (occurs 2 times)

We can arrange the odd digits in the 4 even positions using the formula for permutations of multiset:
\[ \text{Arrangement of odd digits} = \frac{4!}{2! \times 2!} \]
Calculating this gives:
\[ \frac{4!}{2! \times 2!} = \frac{24}{2 \times 2} = 6 \]

### Step 4: Arrange the even digits in the odd positions
The even digits we have are:
- 2 (occurs 2 times)
- 8 (occurs 3 times)

We can arrange the even digits in the 5 odd positions using the formula for permutations of multiset:
\[ \text{Arrangement of even digits} = \frac{5!}{2! \times 3!} \]
Calculating this gives:
\[ \frac{5!}{2! \times 3!} = \frac{120}{2 \times 6} = 10 \]

### Step 5: Calculate the total arrangements
To find the total number of different nine-digit numbers, we multiply the arrangements of odd digits by the arrangements of even digits:
\[ \text{Total arrangements} = 6 \times 10 = 60 \]

### Conclusion
Thus, the total number of different nine-digit numbers that can be formed with the digits 223355888, ensuring that the odd digits occupy even positions, is **60**.

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