The number of arrangements that can be made with the letters of the word 'MATHEMATICS' in which all vowels comes together, is

Options

A. `(8!xx4!)/(2!2!)`
B. `(8!xx4!)/(2!2!2!)`
C. `(8!)/(2!2!2!)`
D. `(8!)/(4!2!2!)`

Explanation

To solve the problem of finding the number of arrangements of the letters in the word "MATHEMATICS" such that all vowels come together, we can follow these steps:

### Step 1: Identify the letters and count them
The word "MATHEMATICS" consists of the following letters:
- M, A, T, H, E, M, A, T, I, C, S

Counting these letters, we find that there are a total of 11 letters.

### Step 2: Identify the vowels and consonants
The vowels in "MATHEMATICS" are:
- A, E, A, I

Counting the vowels, we have:
- Total vowels = 4 (A, A, E, I)

The consonants are:
- M, T, H, M, T, C, S

Counting the consonants, we have:
- Total consonants = 7 (M, T, H, M, T, C, S)

### Step 3: Treat all vowels as a single unit
Since we want all vowels to come together, we can treat the group of vowels (A, A, E, I) as a single unit. Therefore, we can represent the arrangement as:
- [Vowels], M, T, H, M, T, C, S

This gives us a total of 8 units to arrange (1 vowel unit + 7 consonants).

### Step 4: Calculate the arrangements of the units
The total number of arrangements of these 8 units is calculated using the formula for permutations of a multiset:
\[ \text{Total arrangements} = \frac{8!}{2! \times 2!} \]
Here, we divide by \(2!\) for the two M's and \(2!\) for the two T's.

### Step 5: Calculate the arrangements of the vowels
Next, we need to arrange the vowels within their unit. The vowels A, A, E, I can be arranged as:
\[ \text{Arrangements of vowels} = \frac{4!}{2!} \]
We divide by \(2!\) for the two A's.

### Step 6: Combine the arrangements
Now, we multiply the arrangements of the units by the arrangements of the vowels:
\[ \text{Total arrangements with vowels together} = \left(\frac{8!}{2! \times 2!}\right) \times \left(\frac{4!}{2!}\right) \]

### Step 7: Calculate the final answer
Now we can compute the values:
- \(8! = 40320\)
- \(4! = 24\)
- \(2! = 2\)

Substituting these values:
\[ \text{Total arrangements} = \left(\frac{40320}{2 \times 2}\right) \times \left(\frac{24}{2}\right) = \left(\frac{40320}{4}\right) \times 12 = 10080 \times 12 = 120960 \]

Thus, the total number of arrangements of the letters in "MATHEMATICS" such that all vowels come together is **120960**.