In how many ways can the letters of the word ' COMBINE ' be arranged so that,
all the vowels never come together,

Answer

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

### Step 1: Count the total letters and vowels
The word "COMBINE" has 7 letters in total. The vowels in the word are O, I, and E, which gives us 3 vowels.

### Step 2: Calculate the total arrangements of the letters
The total arrangements of the letters in "COMBINE" can be calculated using the factorial of the number of letters:
\[ \text{Total arrangements} = 7! = 5040 \]

### Step 3: Calculate arrangements where vowels come together
To find the arrangements where all the vowels come together, we can treat the vowels (O, I, E) as a single unit or block. This means we will have the following blocks to arrange:
- The block of vowels (OIE)
- The consonants C, M, B, N

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

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

Within the vowel block (OIE), the vowels can be arranged among themselves in:
\[ 3! = 6 \]

Thus, the total arrangements where the vowels come together is:
\[ 5! \times 3! = 120 \times 6 = 720 \]

### Step 4: Calculate arrangements where vowels do not come together
To find the arrangements where the vowels do not come together, we subtract the arrangements where the vowels come together from the total arrangements:
\[ \text{Arrangements where vowels do not come together} = 7! - (5! \times 3!) \]
\[ = 5040 - 720 = 4320 \]

### Final Answer
Thus, the number of ways the letters of the word "COMBINE" can be arranged such that all the vowels never come together is:
\[ \boxed{4320} \]

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