In how many different ways can the letters of word 'IMPLEMENT' be arranged ?

Options

A. 362880
B. 22680
C. 45360
D. 90720

Answer

To find the number of different ways the letters of the word "IMPLEMENT" can be arranged, we can follow these steps:

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

### Step 2: Identify the frequency of each letter
In the word "IMPLEMENT":
- I appears 1 time
- M appears 2 times
- P appears 1 time
- L appears 1 time
- E appears 2 times
- N appears 1 time
- T appears 1 time

### Step 3: Use the formula for permutations of multiset
The formula to calculate the number of arrangements of letters in a word where some letters are repeated is given by:

\[ \text{Number of arrangements} = \frac{n!}{p_1! \times p_2! \times ... \times p_k!} \]

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

For "IMPLEMENT":
- Total letters \( n = 9 \)
- The frequencies are:
- M: 2 times
- E: 2 times

Thus, the formula becomes:

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

### Step 4: Calculate \( 9! \)
Calculating \( 9! \):

\[ 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362880 \]

### Step 5: Calculate \( 2! \)
Calculating \( 2! \):

\[ 2! = 2 \times 1 = 2 \]

### Step 6: Substitute values into the formula
Now substituting the values into the formula:

\[ \text{Number of arrangements} = \frac{362880}{2 \times 2} = \frac{362880}{4} = 90720 \]

### Final Answer
The total number of different ways the letters of the word "IMPLEMENT" can be arranged is **90,720**.

---