Question

The number of words that can be formed using all the letters of the word "DAUGHTER" such that all the vowels never come together, is:

• A. \( 34000 \)
• B. \( 37000 \)
• C. \( 36000 \)
• D. \( 35000 \)

Hint:

When counting arrangements where certain items must not be together, first count the total arrangements, then subtract the unwanted cases (where the items are together).

Answer 1

The Correct Option is C

To solve the problem of finding the number of words that can be formed using all the letters of the word "DAUGHTER" such that all the vowels never come together, we need to follow these steps: 

1. Identify the letters in the word "DAUGHTER". They are: D, A, U, G, H, T, E, R. There are a total of 8 letters.
2. Count the vowels in "DAUGHTER". The vowels are A, U, E. There are 3 vowels.
3. Calculate the total number of arrangements of the 8 letters:

\(8! = 40320\)

1. Calculate the number of arrangements where the vowels all come together. Here, consider the 3 vowels (A, U, E) as one single entity or "super letter". This gives us new entities: (AUE), D, G, H, T, R.
2. Count the number of these entities, which are 6 in number (5 consonants + 1 super letter of vowels).
3. Calculate the arrangements of these 6 entities:

\(6! = 720\)

1. Within the super letter (AUE), the vowels can be arranged among themselves. The number of such arrangements is:

\(3! = 6\)

1. Thus, the total number of arrangements where all vowels come together is:

\(6! \times 3! = 720 \times 6 = 4320\)

1. The required number of arrangements where vowels do not all come together is the total arrangements minus those where vowels are together:

\(8! - 6! \times 3! = 40320 - 4320 = 36000\)

1. Thus, the number of words that can be formed such that all vowels never come together is 36,000.

Therefore, the correct answer is 36000.

Answer 2

Step 1: Total number of words. The total number of words that can be formed using all the letters of the word "DAUGHTER" is: \[ \text{Total words} = 8! = 40320. \] Step 2: Words with all vowels together. The vowels in "DAUGHTER" are A, U, and E. If we treat these vowels as a single entity, the number of arrangements of the letters is: \[ \text{Words with vowels together} = 6! \times 3! = 720 \times 6 = 4320. \] Step 3: Words with vowels not together. To find the number of words where the vowels are not together, subtract the number of words with vowels together from the total number of words: \[ \text{Words with vowels not together} = 8! - 6! \times 3! = 40320 - 4320 = 36000. \]