In how many different ways can the letters of the word ADJUST be arranged so that the vowels never come together ?

Options

A. 720
B. 120
C. 240
D. 480

Answer

To solve the problem of arranging the letters of the word "ADJUST" such that the vowels never come together, we can follow these steps:

### Step 1: Identify the letters and vowels
The word "ADJUST" consists of 6 letters: A, D, J, U, S, T. Among these, the vowels are A and U.

### Step 2: Calculate the total arrangements without any restrictions
To find the total number of arrangements of the letters in "ADJUST", we can use the factorial of the number of letters:
\[ \text{Total arrangements} = 6! = 720 \]

### Step 3: Calculate arrangements where the vowels are together
To find the arrangements where the vowels A and U are together, we can treat the pair (AU) as a single unit. Thus, we can consider the units as follows: (AU), D, J, S, T. This gives us a total of 5 units to arrange.

The number of arrangements of these 5 units is:
\[ 5! = 120 \]

However, since A and U can be arranged among themselves in 2 ways (AU or UA), we need to multiply the arrangements by 2:
\[ \text{Arrangements with vowels together} = 5! \times 2! = 120 \times 2 = 240 \]

### Step 4: Calculate arrangements where the vowels are never together
To find the arrangements where the vowels are never together, we subtract the arrangements where the vowels are together from the total arrangements:
\[ \text{Arrangements with vowels never together} = \text{Total arrangements} - \text{Arrangements with vowels together} \]
\[ = 720 - 240 = 480 \]

### Final Answer
Thus, the number of different ways the letters of the word "ADJUST" can be arranged such that the vowels never come together is:
\[ \boxed{480} \]