How many numbers can be formed using the digits 1, 2, 3, 4, 3, 2, 1 so that the odd digits always occupy the odd places?

Explanation

To solve the problem of how many numbers can be formed using the digits 1, 2, 3, 4, 3, 2, 1 such that the odd digits always occupy the odd places, we will follow these steps:

### Step 1: Identify the digits and their counts
We have the digits: 1, 2, 3, 4, 3, 2, 1.
- Odd digits: 1, 3, 3, 1 (Total: 4 odd digits)
- Even digits: 2, 2, 4 (Total: 3 even digits)

### Step 2: Determine the positions
...