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 find the number of arrangements of the letters in the word "MATHEMATICS" where all vowels come together, we can follow these steps:

### Step 1: Identify the vowels and consonants
The word "MATHEMATICS" consists of the following letters:
- Vowels: A, A, E, I (4 vowels)
- Consonants: M, T, H, M, T, C, S (7 consonants)

### Step 2: 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 or letter. Therefore, we will have:
- Vowel group: (AAEI)
- Consonants: M, T, H, M, T, C, S

This gives us a total of 8 "letters" to arrange:
- 1 vowel group + 7 consonants = 8 units

### Step 3: Calculate the arrangements of the 8 units
The total arrangements of these 8 units (considering that M and T are repeated) can be calculated using the formula for permutations of multiset:
\[ \text{Arrangements} = \frac{8!}{2! \times 2!} \]
Where:
- \(8!\) is the factorial of the total units (8)
- \(2!\) accounts for the two M's
- \(2!\) accounts for the two T's

### Step 4: Calculate the arrangements of the vowels
Next, we need to arrange the vowels within their group (AAEI). The arrangement of these vowels can be calculated as:
\[ \text{Vowel arrangements} = \frac{4!}{2!} \]
Where:
- \(4!\) is the factorial of the total vowels (4)
- \(2!\) accounts for the two A's

### Step 5: Calculate the total arrangements
Now, we multiply the arrangements of the 8 units by the arrangements of the vowels:
\[ \text{Total arrangements} = \left(\frac{8!}{2! \times 2!}\right) \times \left(\frac{4!}{2!}\right) \]

### Step 6: Substitute the factorial values and calculate
Calculating the factorials:
- \(8! = 40320\)
- \(4! = 24\)
- \(2! = 2\)

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

### Final Answer
The total number of arrangements of the letters in "MATHEMATICS" where all vowels come together is **120960**.

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