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. a. 216
B. b. 45360
C. c. 1260
D. d. 43200

Answer

To solve the problem of how many arrangements can be made out of the letters of 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" consists of 9 letters where:
- C = 1
- O = 1
- M = 2
- I = 1
- T = 2
- E = 2

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

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

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

So, we calculate:

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

Calculating this gives:

\[ 9! = 362880 \]
\[ 2! = 2 \quad \text{(for M)} \]
\[ 2! = 2 \quad \text{(for T)} \]
\[ 2! = 2 \quad \text{(for E)} \]

Thus,

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

### Step 2: Count the arrangements where the vowels come together.

The vowels in "COMMITTEE" are O, I, E, E. We can treat these vowels as a single unit or block.

Now, if we consider the block of vowels as a single letter, we have the following letters to arrange:
- Vowel block (OIEE)
- C
- M
- M
- T
- T

This gives us a total of 6 letters to arrange (the vowel block + C + M + M + T + T).

The arrangements of these 6 letters (considering repetitions) is given by:

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

Calculating this gives:

\[ 6! = 720 \]
\[ 2! = 2 \quad \text{(for M)} \]
\[ 2! = 2 \quad \text{(for T)} \]

Thus,

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

### Step 3: Count the arrangements of the vowels within the block.

The vowels O, I, E, E can be arranged among themselves. The arrangements of these 4 letters (considering repetitions) is given by:

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

Calculating this gives:

\[ 4! = 24 \]
\[ 2! = 2 \quad \text{(for E)} \]

Thus,

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

### Step 4: Total arrangements where vowels are together.

Now, we multiply the arrangements of the letters with the arrangements of the vowels:

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

### Step 5: Calculate the arrangements where the 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:

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

Calculating this gives:

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

### Final Answer:

Thus, 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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