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 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: Calculate the total arrangements without restrictions
1. Identify the letters in "COMMITTEE":
- C, O, M, M, I, T, T, E, E
2. Count the total letters:
- There are 9 letters in total.
3. Identify the repeating letters:
- M appears 2 times, T appears 2 times, and E appears 2 times.
4. Use the formula for permutations of multiset:
\[ \text{Total arrangements} = \frac{n!}{p_1! \cdot p_2! \cdot \ldots \cdot p_k!} \]
where \( n \) is the total number of letters and \( p_i \) are the counts of each repeating letter.
5. Substitute the values:
\[ \text{Total arrangements} = \frac{9!}{2! \cdot 2! \cdot 2!} \]
6. Calculate:
\[ 9! = 362880, \quad 2! = 2 \quad \text{(for M, T, and E)} \]
\[ \text{Total arrangements} = \frac{362880}{2 \cdot 2 \cdot 2} = \frac{362880}{8} = 45360 \]

### Step 2: Calculate arrangements where vowels are together
1. Identify the vowels in "COMMITTEE":
- O, I, E, E (4 vowels)
2. Treat the vowels as a single unit:
- This gives us the units: (OIEE), C, M, M, T, T → 5 units in total.
3. Count the arrangements of these units:
\[ \text{Arrangements with vowels together} = \frac{5!}{2! \cdot 2!} \]
4. Calculate:
\[ 5! = 120, \quad 2! = 2 \quad \text{(for M and T)} \]
\[ \text{Arrangements with vowels together} = \frac{120}{2 \cdot 2} = \frac{120}{4} = 30 \]
5. Now, arrange the vowels (O, I, E, E):
\[ \text{Arrangements of vowels} = \frac{4!}{2!} = \frac{24}{2} = 12 \]
6. Total arrangements with vowels together:
\[ \text{Total} = 30 \cdot 12 = 360 \]

### Step 3: Calculate arrangements where vowels do not come together
1. Subtract the arrangements where vowels are together from the total arrangements:
\[ \text{Arrangements where vowels do not come together} = 45360 - 360 = 44940 \]

### Final Answer
The number of arrangements of the letters of the word "COMMITTEE" such that the four vowels do not come together is **44940**.