In how many ways can the letters of the word ' COMBINE ' be arranged so that, the vowels are never separated

Answer

To find the number of ways to arrange the letters of the word "COMBINE" such that the vowels are never separated, we can follow these steps:

### Step 1: Identify the letters in the word "COMBINE"
The word "COMBINE" consists of 7 letters: C, O, M, B, I, N, E.

### Step 2: Identify the vowels and consonants
In "COMBINE", the vowels are O, I, and E. The consonants are C, M, B, and N.
- Vowels: O, I, E (3 vowels)
- Consonants: C, M, B, N (4 consonants)

### Step 3: Treat the vowels as a single unit
Since we want to ensure that the vowels are never separated, we can treat the group of vowels (OIE) as a single unit or block. Therefore, instead of 7 individual letters, we now have:
- 1 block of vowels (OIE)
- 4 consonants (C, M, B, N)

This gives us a total of 5 units to arrange: (OIE), C, M, B, N.

### Step 4: Calculate the arrangements of the 5 units
The number of ways to arrange these 5 units is given by the factorial of the number of units:
\[ 5! = 120 \]

### Step 5: Calculate the arrangements of the vowels within their block
The vowels O, I, and E can be arranged among themselves. The number of ways to arrange these 3 vowels is:
\[ 3! = 6 \]

### Step 6: Calculate the total arrangements
To find the total number of arrangements where the vowels are never separated, we multiply the number of arrangements of the 5 units by the number of arrangements of the vowels:
\[ \text{Total arrangements} = 5! \times 3! = 120 \times 6 = 720 \]

### Final Answer
Thus, the total number of ways to arrange the letters of the word "COMBINE" such that the vowels are never separated is **720**.

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