Consider the Word W=TERRORIST.
Q. Number of arrangements of the word W, if R's as well as T's are separated, is

Options

A. 9860
B. 1080
C. 10200
D. 11400

Explanation

To find the number of arrangements of the word "TERRORIST" such that the R's and T's are separated, we can follow these steps:

### Step 1: Count the total letters and their frequencies
The word "TERRORIST" consists of 9 letters with the following frequencies:
- T: 2
- R: 3
- E: 1
- O: 1
- I: 1
- S: 1

### Step 2: Calculate the total arrangements without restrictions
The total number of arrangements of the letters in "TERRORIST" can be calculated using the formula for permutations of multiset:

\[ \text{Total arrangements} = \frac{n!}{n_1! \times n_2! \times n_3! \times \ldots} \]

Where \( n \) is the total number of letters, and \( n_1, n_2, n_3, \ldots \) are the frequencies of each distinct letter.

Here, \( n = 9 \), \( n_T = 2 \), \( n_R = 3 \), \( n_E = 1 \), \( n_O = 1 \), \( n_I = 1 \), \( n_S = 1 \).

So, the total arrangements are:

\[ \text{Total arrangements} = \frac{9!}{2! \times 3! \times 1! \times 1! \times 1! \times 1!} \]

Calculating this gives:

\[ 9! = 362880 \]
\[ 2! = 2, \quad 3! = 6 \]
\[ \text{Total arrangements} = \frac{362880}{2 \times 6 \times 1 \times 1 \times 1 \times 1} = \frac{362880}{12} = 30240 \]

### Step 3: Calculate arrangements where R's and T's are together
To find the arrangements where R's and T's are not separated, we can treat the groups of R's and T's as single units.

- Treat the 3 R's as a single unit (RRR) and the 2 T's as another single unit (TT).
- This gives us the units: RRR, TT, E, O, I, S which totals to 6 units.

Now we can calculate the arrangements of these 6 units:

\[ \text{Arrangements with R's and T's together} = \frac{6!}{1! \times 1! \times 1! \times 1! \times 1!} = 6! = 720 \]

However, we also need to account for the arrangements within the R's and T's:

- The arrangements of R's within RRR is 1 (since they are identical).
- The arrangements of T's within TT is also 1.

Thus, the total arrangements where R's and T's are together is:

\[ \text{Total arrangements with R's and T's together} = 720 \times 1 \times 1 = 720 \]

### Step 4: Calculate arrangements where R's and T's are separated
Now, we can find the arrangements where R's and T's are separated by subtracting the arrangements where they are together from the total arrangements:

\[ \text{Arrangements where R's and T's are separated} = \text{Total arrangements} - \text{Arrangements with R's and T's together} \]

\[ = 30240 - 720 = 29520 \]

### Final Answer
The number of arrangements of the word "TERRORIST" such that the R's and T's are separated is **29520**.

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