Question

Find the number of different 8-letter arrangements that can be made from the letters of the word daughter so that

all vowels occur together

Answer 1

To find the number of 8-letter arrangements from the word DAUGHTER with vowels together, treat vowels as one unit and find arrangements for 6 units (6!) and the arrangements within the vowel unit (3!), then multiply these values to get 4320 different arrangements.

Step-by-step explanation:

You asked about finding the number of different 8-letter arrangements that can be made from the letters of the word DAUGHTER so that all vowels occur together. Let's break down the steps to solve this:

1. Identify the vowels in the word DAUGHTER, which are A, U, and E.
2. Treat these vowels as a single entity since they must occur together. So, you will be arranging the entities D, G, H, T, R, and (AUE).
3. Calculate arrangements of these 6 entities. This can be done in 6! (factorial) ways.
4. Now consider the different arrangements of the vowels A, U, and E within their own group. This can be done in 3! ways.
5. To find the total arrangements, multiply the number of arrangements of the 6 entities by the number of arrangements of the vowels: 6! x 3!.
6. Compute the factorials: 6! = 720 and 3! = 6, giving us 720 x 6 = 4320 different arrangements where all vowels occur together.

Thus, there are 4320 different arrangements where the vowels in the word DAUGHTER occur together.