Understanding the Problem
To arrange the letters of the word "DAUGHTER" with vowels in odd positions, we first identify the vowels and consonants:
- Vowels: A, U, E (3 vowels)
- Consonants: D, G, H, T, R (5 consonants)
Vowel Placement in Odd Positions
The odd positions in the word "DAUGHTER" (which has 8 letters) are:
- 1st, 3rd, 5th, and 7th positions (4 odd positions)
However, since we only have 3 vowels, we will fill 3 odd positions out of the available 4.
Selecting Vowel Positions
- We need to choose 3 positions from the 4 available odd positions. This can be calculated using combinations:
- Number of ways to choose 3 positions from 4 = 4C3 = 4.
Arranging the Vowels
- The 3 vowels (A, U, E) can be arranged among themselves in:
- 3! = 6 ways.
Arranging the Consonants
- The remaining 5 consonants (D, G, H, T, R) can fill the remaining 5 positions (including the unoccupied odd position) in:
- 5! = 120 ways.
Calculating the Total Arrangements
- Total arrangements can be found by multiplying the number of ways to choose vowel positions, the arrangements of vowels, and the arrangements of consonants:
- Total = (4C3) * (3!) * (5!)
= 4 * 6 * 120
= 2880.
Final Answer
Thus, the total number of different arrangements of the letters in "DAUGHTER" with vowels occupying the odd places is 2880.