How many arrangements can be made out of the letters of the word COMMITTEE , taken all at a time , such that the four vowels do not come together ?

Options

A. 12600
B. 45600
C. 43200
D. 216

Answer

To find the number of arrangements of the letters in the word "COMMITTEE" such that the four vowels do not come together, we can follow these steps:

### Step 1: Count the total arrangements of the letters in "COMMITTEE".

The word "COMMITTEE" has 9 letters in total: C, O, M, M, I, T, T, E, E.

Among these:
- M appears 2 times
- T appears 2 times
- E appears 2 times
- C, O, and I appear 1 time each

The formula for the total arrangements of letters when there are repetitions is given by:

\[ \text{Total arrangements} = \frac{n!}{p_1! \times p_2! \times p_3! \ldots} \]

Where \( n \) is the total number of letters, and \( p_1, p_2, p_3, \ldots \) are the frequencies of the repeated letters.

Thus, the total arrangements are:

\[ \text{Total arrangements} = \frac{9!}{2! \times 2! \times 2!} \]

### Step 2: Calculate the total arrangements.

Calculating \( 9! \):

\[ 9! = 362880 \]

Calculating \( 2! \):

\[ 2! = 2 \]

So, the total arrangements become:

\[ \text{Total arrangements} = \frac{362880}{2 \times 2 \times 2} = \frac{362880}{8} = 45360 \]

### Step 3: Count the arrangements where all vowels are together.

The vowels in "COMMITTEE" are O, I, E, E.

We can treat the group of vowels as a single entity or block. Thus, we have the following letters to arrange:

- The block of vowels (OIEE)
- C, M, M, T, T

This gives us 6 entities to arrange: (OIEE), C, M, M, T, T.

The number of arrangements of these 6 entities is:

\[ \text{Arrangements with vowels together} = \frac{6!}{2! \times 2!} \]

Calculating \( 6! \):

\[ 6! = 720 \]

So, the arrangements become:

\[ \text{Arrangements with vowels together} = \frac{720}{2 \times 2} = \frac{720}{4} = 180 \]

### Step 4: Count the arrangements of the vowels within the block.

The vowels O, I, E, E can be arranged among themselves as:

\[ \text{Arrangements of vowels} = \frac{4!}{2!} \]

Calculating \( 4! \):

\[ 4! = 24 \]

So, the arrangements of the vowels become:

\[ \text{Arrangements of vowels} = \frac{24}{2} = 12 \]

### Step 5: Calculate total arrangements with vowels together.

Now, we multiply the arrangements of the entities by the arrangements of the vowels:

\[ \text{Total arrangements with vowels together} = 180 \times 12 = 2160 \]

### Step 6: Calculate the arrangements where vowels do not come together.

To find the arrangements where the vowels do not come together, we subtract the arrangements where they do come together from the total arrangements:

\[ \text{Arrangements where vowels do not come together} = \text{Total arrangements} - \text{Total arrangements with vowels together} \]

Calculating this gives:

\[ \text{Arrangements where vowels do not come together} = 45360 - 2160 = 43100 \]

### Final Answer:

The number of arrangements of the letters of the word "COMMITTEE" such that the four vowels do not come together is **43100**.

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