Consider the word `W = TER RORIST` Number of arrangements of the word w, if no two R's are together, is

Options

A. 11460
B. 10400
C. 12600
D. 9860

Explanation

To find the number of arrangements of the word "TERRORIST" such that no two R's are together, we can follow these steps:

### Step 1: Count the total letters and identify duplicates
The word "TERRORIST" consists of 9 letters: T, E, R, R, O, R, I, S, T.
- The letters are: T, E, R (3 times), O, I, S.
- The total number of letters = 9.
- The duplicates are: R appears 3 times and T appears 2 times.

### Step 2: Arrange the letters excluding R
First, we will arrange the letters excluding R. The letters we have are: T, E, O, I, S.
- Total distinct letters = 5 (T, E, O, I, S).

The total arrangements of these 5 letters is given by:
\[ \text{Arrangements} = 5! = 120 \]

### Step 3: Identify positions for R's
When we arrange the letters T, E, O, I, S, we create gaps where we can place the R's. The arrangement of T, E, O, I, S will create 6 gaps (before the first letter, between the letters, and after the last letter):
- _ T _ E _ O _ I _ S _

Thus, we have 6 gaps to place the R's.

### Step 4: Choose gaps for R's
We need to choose 3 out of these 6 gaps to place the R's. The number of ways to choose 3 gaps from 6 is given by:
\[ \text{Ways to choose gaps} = \binom{6}{3} \]

Calculating this:
\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

### Step 5: Calculate total arrangements
Now, we multiply the number of arrangements of T, E, O, I, S by the number of ways to place the R's:
\[ \text{Total arrangements} = \text{Arrangements of T, E, O, I, S} \times \text{Ways to choose gaps} \]
\[ \text{Total arrangements} = 120 \times 20 = 2400 \]

### Final Answer
The total number of arrangements of the word "TERRORIST" such that no two R's are together is **2400**.

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