Number of different words that can be made using the letters of the word HALLUCINATION if all the consonants are together is

Options

A. `((7!)/(2!2!))^(2)`
B. `(7!)/(2!2!)`
C. `(14!)/(7!)`
D. `(14)/(2!2!)`

Answer

To solve the problem of finding the number of different words that can be made using the letters of the word "HALLUCINATION" with the condition that all consonants are together, we can follow these steps:

### Step 1: Identify the letters in "HALLUCINATION"
The word "HALLUCINATION" consists of 13 letters:
- Consonants: H, L, L, C, N, T, N (7 consonants)
- Vowels: A, U, I, A, I, O (6 vowels)

### Step 2: Count the repeated letters
- Consonants:
- L appears 2 times
- N appears 2 times
- Vowels:
- A appears 2 times
- I appears 2 times

### Step 3: Treat all consonants as a single unit
Since we need to keep all the consonants together, we can treat them as one single unit or block. This means we can represent the consonants as [LLCNTN].

### Step 4: Count the total units
Now, we have:
- 1 block of consonants
- 6 vowels (A, U, I, A, I, O)

This gives us a total of 1 (consonant block) + 6 (vowels) = 7 units to arrange.

### Step 5: Arrange the units
The arrangement of these 7 units (1 consonant block + 6 vowels) can be calculated as follows:
- Total arrangements = 7! / (2! * 2!)
- The 2! accounts for the two A's and the other 2! accounts for the two I's.

### Step 6: Arrange the consonants within the block
Now, we need to arrange the consonants within the block [LLCNTN]. The arrangement of these consonants can be calculated as:
- Total arrangements = 7! / (2! * 2!)
- The 2! accounts for the two L's and the other 2! accounts for the two N's.

### Step 7: Combine the arrangements
The total number of different words where all consonants are together is given by:
- Total arrangements = (Arrangements of units) * (Arrangements of consonants)
- Total arrangements = (7! / (2! * 2!)) * (7! / (2! * 2!))

### Step 8: Calculate the final answer
Now we can calculate the numerical value:
- 7! = 5040
- 2! = 2

Thus, the total number of arrangements is:
- Total = (5040 / (2 * 2)) * (5040 / (2 * 2))
- Total = (5040 / 4) * (5040 / 4)
- Total = 1260 * 1260 = 1587600

### Final Answer
The total number of different words that can be formed from the letters of "HALLUCINATION" with all consonants together is **1587600**.

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