In how many different ways can the word 'COFFEE' can be arranged so as vowel always come together?

Answer

To solve the problem of arranging the word "COFFEE" such that the vowels always come together, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Vowels and Consonants:**
- The word "COFFEE" consists of the vowels: O, E, E (total 3 vowels).
- The consonants are: C, F, F (total 3 consonants).

2. **Group the Vowels Together:**
- Since the vowels must always come together, we can treat the group of vowels (OEE) as a single unit.
- Thus, we can represent the arrangement as: (OEE), C, F, F.

3. **Count the Total Units:**
- Now we have 4 units to arrange: (OEE), C, F, F.

4. **Calculate the Arrangements of the Units:**
- The total arrangements of these 4 units can be calculated using the formula for permutations of multiset:
\[ \text{Arrangements} = \frac{n!}{p_1! \times p_2!} \]
- Here, \( n = 4 \) (total units), and we have 2 F's which are repeating.
- Therefore, the number of arrangements is:
\[ \frac{4!}{2!} = \frac{24}{2} = 12 \]

5. **Arrange the Vowels:**
- Next, we need to arrange the vowels within their group (OEE).
- The number of arrangements of the vowels is:
\[ \frac{3!}{2!} = \frac{6}{2} = 3 \]
- (Here, we divide by 2! because the letter E is repeated).

6. **Calculate the Total Arrangements:**
- Finally, we multiply the arrangements of the units by the arrangements of the vowels:
\[ \text{Total Arrangements} = 12 \times 3 = 36 \]

### Final Answer:
The total number of different ways to arrange the word "COFFEE" such that the vowels always come together is **36**.

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