Introduction
The given word is ORIENTAL.

Step 1: Fixing the positions of A and E
As per the given condition, A and E should always occupy odd places. Hence, we can fix their positions in the following ways:
- AE------
- -AE-----
- --AE----
- ----AE-
- -----AE

Step 2: Filling the remaining positions
The remaining positions can be filled with the remaining letters of the word ORIENTAL. As we have already fixed the positions of A and E, we can only use the letters O, R, I, N, T, and L to fill the remaining positions.

Step 3: Calculation of permutations
We can use the formula for permutations to calculate the number of words that can be formed:
- Number of ways to fill the first position = 6 (since we can use any of the 6 remaining letters)
- Number of ways to fill the second position = 5 (since we have used one letter already)
- Number of ways to fill the third position = 4
- Number of ways to fill the fourth position = 3
- Number of ways to fill the fifth position = 2
- Number of ways to fill the sixth position = 1

Hence, the total number of words that can be formed with the letters of the word ORIENTAL so that A and E always occupy odd places is:
6 x 5 x 4 x 3 x 2 x 1 = 720

Conclusion
Therefore, there are 720 words that can be formed with the letters of the word ORIENTAL so that A and E always occupy odd places.