Final answer:
The letters of the word "Daughters" can be arranged in 4320 different ways while keeping the vowels together by multiplying 6! (for all entities including the vowel group) and 3! (for arrangements of vowels within the entity).
Step-by-step explanation:
To determine in how many ways the letters of the word "Daughters" can be arranged keeping the vowels together, we consider the vowels (A, U, E) as a single entity. Thus, we have D, G, H, T, R, and the entity (AUE) to arrange. We need to find the factorial of the number of entities, which is 6!, to find all arrangements of these entities.
However, the vowels (A, U, E) within the entity can also be arranged among themselves in several ways, specifically 3! ways. Therefore, to find the total number of arrangements with vowels together, we need to multiply these two numbers:
- Arrangements of all entities = 6!
- Arrangements of vowels within the entity = 3!
The total number of arrangements is therefore 6! * 3!, which equals to 720 * 6 = 4320 different ways.