In how many different ways can the word 'HOME' be arranged so as, the vowels never come together?

Answer

To solve the problem of arranging the word "HOME" such that the vowels (O and E) never come together, we can follow these steps:

### Step 1: Identify the vowels and consonants
The word "HOME" consists of 4 letters:
- Vowels: O, E
- Consonants: H, M

### Step 2: Calculate the total arrangements of the letters
First, we calculate the total arrangements of the letters in "HOME" without any restrictions. Since all letters are distinct, the total arrangements can be calculated using the factorial of the number of letters.

Total arrangements = 4! = 24

### Step 3: Calculate arrangements where vowels are together
To find the arrangements where the vowels are together, we can treat the vowels (O and E) as a single unit or block. Therefore, we can consider the block (OE) along with the consonants H and M.

Now, we have three units to arrange: (OE), H, M.

The arrangements of these three units can be calculated as:
Arrangements = 3! = 6

Within the block (OE), the vowels can be arranged among themselves in:
Arrangements = 2! = 2

Thus, the total arrangements where the vowels are together is:
Total arrangements with vowels together = 3! * 2! = 6 * 2 = 12

### Step 4: Calculate arrangements where vowels are NOT together
To find the arrangements where the vowels are not together, we subtract the arrangements where the vowels are together from the total arrangements.

Arrangements where vowels are NOT together = Total arrangements - Arrangements with vowels together
= 24 - 12 = 12

### Final Answer
The number of different ways the word "HOME" can be arranged such that the vowels never come together is **12**.

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