The number of words that can be formed out of the letters of the word COMMITTEE is

Options

A. `(9!)/((2!)^3)`
B. `(9!)/((2!)^2)`
C. `(9!)/(2!)`
D. `9!`

Answer

To find the number of distinct words that can be formed from the letters of the word "COMMITTEE", we will follow these steps:

### Step 1: Count the total number of letters
The word "COMMITTEE" consists of 9 letters.

### Step 2: Identify the frequency of each letter
In the word "COMMITTEE", the frequency of each letter is as follows:
- C: 1
- O: 1
- M: 2
- I: 1
- T: 2
- E: 2

### Step 3: Use the formula for permutations of multiset
The formula to find the number of distinct permutations of letters in a word is given by:

\[ \text{Number of permutations} = \frac{n!}{p_1! \times p_2! \times p_3! \times \ldots} \]

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

### Step 4: Substitute the values into the formula
Here, \( n = 9 \) (total letters), and the repeated letters are:
- M: 2 times
- T: 2 times
- E: 2 times

Thus, we have:

\[ \text{Number of permutations} = \frac{9!}{2! \times 2! \times 2!} \]

### Step 5: Calculate the factorials
Now we calculate \( 9! \) and \( 2! \):

\[ 9! = 362880 \]
\[ 2! = 2 \]

### Step 6: Substitute the factorial values into the equation
Now substituting back into the equation:

\[ \text{Number of permutations} = \frac{362880}{2 \times 2 \times 2} = \frac{362880}{8} \]

### Step 7: Perform the division
Now, we calculate:

\[ \frac{362880}{8} = 45360 \]

### Conclusion
The total number of distinct words that can be formed from the letters of the word "COMMITTEE" is **45360**.

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