Find how many words can be formed from the word COMBINATION ,such that vowels will always come together .

Options

A. 78200
B. 75600
C. 64800
D. 52600

Answer

To solve the problem of finding how many words can be formed from the word "COMBINATION" such that the vowels always come together, we can follow these steps:

### Step 1: Identify the vowels and consonants in the word "COMBINATION"
The word "COMBINATION" consists of the following letters:
- Vowels: O, I, A, I, O (total of 5 vowels)
- Consonants: C, M, B, N, T, N (total of 6 consonants)

### Step 2: Treat the vowels as a single unit
Since we want the vowels to always come together, we can treat the group of vowels (OIAIO) as a single unit. This means we will consider the vowels as one "letter".

### Step 3: Count the total units
Now, we have the following units to arrange:
- Vowel unit (OIAIO)
- C
- M
- B
- N
- T
- N

This gives us a total of 7 units (1 vowel unit + 6 consonants).

### Step 4: Calculate the arrangements of the units
The total arrangements of these 7 units can be calculated using the formula for permutations of n items where some items are identical. In this case, we have two N's among the consonants.

The formula is:
\[ \text{Arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots} \]
Where \( n \) is the total number of units, and \( p_1, p_2, \ldots \) are the counts of identical items.

Here, we have:
- Total units (n) = 7
- Identical consonants (N) = 2

Thus, the arrangements of the units are:
\[ \text{Arrangements} = \frac{7!}{2!} \]

### Step 5: Calculate the arrangements of the vowels
Next, we need to arrange the vowels within their unit. The vowels OIAIO consist of 5 letters where I appears twice. Thus, the arrangements of the vowels are:
\[ \text{Vowel arrangements} = \frac{5!}{2!} \]

### Step 6: Calculate the total arrangements
Finally, to find the total number of arrangements where the vowels are together, we multiply the arrangements of the units by the arrangements of the vowels:
\[ \text{Total arrangements} = \frac{7!}{2!} \times \frac{5!}{2!} \]

### Step 7: Calculate the final answer
Now we can compute the values:
- \( 7! = 5040 \)
- \( 5! = 120 \)
- \( 2! = 2 \)

So,
\[ \text{Total arrangements} = \frac{5040}{2} \times \frac{120}{2} = 2520 \times 60 = 151200 \]

Thus, the total number of words that can be formed from the word "COMBINATION" such that the vowels always come together is **151200**.

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