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

Answer

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

### Step 1: Identify the letters and vowels
The word "COFFEE" consists of the letters: C, O, F, F, E, E.
Here, the vowels are O, E, and E.

### Step 2: Calculate the total arrangements of the letters
The total number of letters in "COFFEE" is 6. However, we have repetitions: F appears twice and E appears twice.

The formula for the total arrangements of letters when there are repetitions is given by:
\[ \text{Total arrangements} = \frac{n!}{p_1! \times p_2!} \]
where \( n \) is the total number of letters, and \( p_1, p_2, \ldots \) are the frequencies of the repeated letters.

So, we calculate:
\[ \text{Total arrangements} = \frac{6!}{2! \times 2!} = \frac{720}{2 \times 2} = \frac{720}{4} = 180 \]

### Step 3: Calculate arrangements where vowels are together
Next, we treat the vowels O, E, E as a single unit or letter. So, we can represent the arrangement as:
C, F, F, (OEE)

Now we have 4 units to arrange: C, F, F, and (OEE).

The arrangements of these 4 units, considering F repeats, is given by:
\[ \text{Arrangements with vowels together} = \frac{4!}{2!} = \frac{24}{2} = 12 \]

### Step 4: Calculate arrangements of the vowels within their unit
Now, we need to arrange the vowels O, E, E within their unit. The arrangements of these vowels, considering E repeats, is given by:
\[ \text{Arrangements of vowels} = \frac{3!}{2!} = \frac{6}{2} = 3 \]

### Step 5: Calculate total arrangements with vowels together
Now, we multiply the arrangements of the units by the arrangements of the vowels:
\[ \text{Total arrangements with vowels together} = 12 \times 3 = 36 \]

### Step 6: Calculate arrangements where vowels never come together
Finally, we subtract the arrangements where the vowels are together from the total arrangements:
\[ \text{Arrangements where vowels never come together} = \text{Total arrangements} - \text{Arrangements with vowels together} \]
\[ = 180 - 36 = 144 \]

### Final Answer
Thus, the number of different ways to arrange the word "COFFEE" such that the vowels never come together is **144**.

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