Introduction:
In this problem, we need to arrange the letters of the word DAUGHTER in such a way that the vowels occupy the odd places.

Method:
To solve this problem, we can follow these steps:

1. First, we need to identify the vowels in the word DAUGHTER, which are A, U, and E.
2. Since we need to arrange the vowels in the odd places, we can start by placing A in the first position, which is an odd place.
3. Next, we can place U in the third position, which is also an odd place.
4. Finally, we can place E in the fifth position, which is the last odd place.
5. Now, we need to arrange the remaining consonants, which are D, G, H, and T, in the even places.
6. We can start by placing D in the second position, which is an even place.
7. Next, we can place G in the fourth position, which is also an even place.
8. Then, we can place H in the sixth position, which is the last even place.
9. Finally, we can place T in the seventh and last position.

Answer:
Therefore, the word DAUGHTER can be arranged in 12 different ways such that the vowels occupy the odd places. These arrangements are:

- A D U G H E T R
- A D U G T H E R
- A D U H G E T R
- A D U H G T E R
- A D U T G H E R
- A D U T H G E R
- D A U G H E T R
- D A U G T H E R
- D A U H G E T R
- D A U H G T E R
- D A U T G H E R
- D A U T H G E R

Therefore, the answer is 12.

Heya!

The number of different ways are "2880" so that vowels occupy odd places.

Explanation:

The Word DAUGHTER has 8 letters

The Number of Vowels in this word is 3

The total odd places available to rearrange are 4

So vowels can be arranged in 4P3 ways which is

= 4 × 3 × 2 = 24 ways

The Number of ways other letter left(8-3) arranged are:
= 5! = 5 × 4 × 3 × 2 × 1 = 120 ways

In total the ways to arrange vowels(at odd places) and other letters of word "DAUGHTER" are =

= 24 × 120 = 2880 ways.

Hope it helps!
All the best!
>GK