Find the number of permutations of the letter of the words
' COMMISIONS.'

Answer

To find the number of permutations of the letters in the word "COMMISIONS," we need to follow these steps:

### Step 1: Count the Total Letters
The word "COMMISIONS" consists of 10 letters.

### Step 2: Identify Repeated Letters
Next, we identify the letters that are repeated:
- C: 1
- O: 2
- M: 2
- I: 2
- S: 2
- N: 1

### Step 3: Use the Permutation Formula
The formula for permutations of a multiset 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,
- \( p_1, p_2, p_3, \ldots \) are the frequencies of the repeated letters.

In our case:
- \( n = 10 \) (total letters)
- \( p_1 = 2 \) (for O)
- \( p_2 = 2 \) (for M)
- \( p_3 = 2 \) (for I)
- \( p_4 = 2 \) (for S)

### Step 4: Substitute Values into the Formula
Now we can substitute these values into the formula:

\[ \text{Number of permutations} = \frac{10!}{2! \times 2! \times 2! \times 2!} \]

### Step 5: Calculate Factorials
Now we calculate the factorials:
- \( 10! = 3628800 \)
- \( 2! = 2 \)

Thus, we have:

\[ \text{Number of permutations} = \frac{3628800}{2 \times 2 \times 2 \times 2} = \frac{3628800}{16} \]

### Step 6: Final Calculation
Now, we perform the final division:

\[ \frac{3628800}{16} = 226800 \]

### Final Answer
The total number of permutations of the letters in the word "COMMISIONS" is **226800**.

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