How many different words can be made using the letters of the word 'HALLUCINATION' if all consonants are together?

Options

A. 129780
B. 1587600
C. 35600
D. none of these

Answer

To solve the problem of how many different words can be made using the letters of the word "HALLUCINATION" with all consonants together, we can follow these steps:

### Step 1: Identify the consonants and vowels
The word "HALLUCINATION" consists of the following letters:
- Consonants: H, L, L, C, N, T, N
- Vowels: A, U, I, A, I, O

### Step 2: Group the consonants together
Since we want all consonants to be together, we can treat the group of consonants (HLLCNTN) as a single unit. This means we can represent the arrangement as:
- [HLLCNTN], A, U, I, A, I, O

### Step 3: Count the total units
Now, we have the following units to arrange:
- 1 unit of consonants
- 5 vowels (A, U, I, A, I, O)

This gives us a total of 6 units to arrange:
- Total units = 1 (consonant group) + 5 (vowels) = 6 units

### Step 4: Calculate arrangements of the units
The number of ways to arrange these 6 units is given by the factorial of the number of units:
- Arrangements of units = 6!

### Step 5: Account for identical letters
However, we have identical letters among the vowels:
- A appears 2 times
- I appears 2 times

Thus, we need to divide by the factorial of the counts of identical letters:
- Arrangements of vowels = 6! / (2! * 2!)

### Step 6: Calculate the arrangements of consonants
Next, we need to arrange the consonants within their group. The consonants are H, L, L, C, N, T, N, which gives us:
- Total consonants = 7 (H, L, L, C, N, T, N)

Again, we have identical letters:
- L appears 2 times
- N appears 2 times

Thus, the arrangements of the consonants is:
- Arrangements of consonants = 7! / (2! * 2!)

### Step 7: Combine the arrangements
The total number of arrangements where all consonants are together is the product of the arrangements of the units and the arrangements of the consonants:
- Total arrangements = (6! / (2! * 2!)) * (7! / (2! * 2!))

### Step 8: Calculate the final answer
Now we can calculate the values:
- 6! = 720
- 7! = 5040
- 2! = 2

Putting it all together:
- Total arrangements = (720 / (2 * 2)) * (5040 / (2 * 2))
- Total arrangements = (720 / 4) * (5040 / 4)
- Total arrangements = 180 * 1260
- Total arrangements = 226800

Thus, the final answer is **226800**.