The number of ways in which the letters of the word PESSIMISTIC can be arranged so that no two S's are together, no of two I's are together and letters `S` and `I` are never together is

Options

A. `8640`
B. `4800`
C. `2400`
D. `5480`

Explanation

To solve the problem of arranging the letters of the word "PESSIMISTIC" such that no two S's are together, no two I's are together, and S's and I's are never together, we can follow these steps:

### Step 1: Identify the letters and their frequencies
The word "PESSIMISTIC" consists of the following letters:
- P: 1
- E: 1
- S: 3
- I: 3
- M: 1
- T: 1
- C: 1

### Step 2: Calculate the total number of letters
The total number of letters in "PESSIMISTIC" is 11.

### Step 3: Arrange the distinct letters
First, we arrange the distinct letters P, E, M, T, C. There are 5 distinct letters.

The number of ways to arrange these 5 letters is:
\[ 5! = 120 \]

### Step 4: Determine the gaps for S's and I's
Once we arrange P, E, M, T, C, we will have 6 gaps (including the ends) to place the S's and I's:
- _ P _ E _ M _ T _ C _

### Step 5: Place S's in the gaps
We need to place 3 S's in these 6 gaps such that no two S's are together. The number of ways to choose 3 gaps from 6 is given by:
\[ \binom{6}{3} = 20 \]

### Step 6: Place I's in the remaining gaps
After placing the S's, we will have 3 gaps left. We need to place 3 I's in these 3 gaps. Since we have to ensure that no two I's are together, we can place one I in each of the remaining gaps. The number of ways to arrange the 3 I's in these 3 gaps is:
\[ 1 \text{ (only one way to place one I in each gap)} \]

### Step 7: Calculate the total arrangements
Now, we can calculate the total arrangements by multiplying the arrangements of distinct letters, the ways to choose gaps for S's, and the arrangements for I's:
\[ \text{Total arrangements} = (5!) \times \binom{6}{3} \times 1 = 120 \times 20 \times 1 = 2400 \]

### Final Answer
The total number of ways to arrange the letters of the word "PESSIMISTIC" such that no two S's are together, no two I's are together, and S's and I's are never together is **2400**.