The number of ways in which letter of the word `'"ARRANGE"'` can be arranged, such that no two R's are together, is

Options

A. 160
B. 200
C. 360
D. 900

Answer

To solve the problem of arranging the letters of the word "ARRANGE" such that no two R's are together, we can follow these steps:

### Step 1: Calculate the total arrangements of the letters in "ARRANGE".
The word "ARRANGE" consists of 7 letters where:
- A appears 2 times,
- R appears 2 times,
- N appears 1 time,
- G appears 1 time,
- E appears 1 time.

The total arrangements can be calculated using the formula for permutations of multiset:

\[ \text{Total arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!} \]

Where \(n\) is the total number of letters, and \(p_i\) are the frequencies of each letter.

\[ \text{Total arrangements} = \frac{7!}{2! \times 2!} = \frac{5040}{4} = 1260 \]

### Step 2: Calculate the arrangements where the R's are together.
To consider the R's as a single entity, we can treat "RR" as one letter. This gives us the letters: "RR", "A", "A", "N", "G", "E", which totals to 6 letters.

Now, we calculate the arrangements of these 6 letters:

\[ \text{Arrangements with R's together} = \frac{6!}{2!} = \frac{720}{2} = 360 \]

### Step 3: Calculate the arrangements where no two R's are together.
To find the arrangements where no two R's are together, we subtract the arrangements where R's are together from the total arrangements:

\[ \text{Arrangements with no R's together} = \text{Total arrangements} - \text{Arrangements with R's together} \]

\[ \text{Arrangements with no R's together} = 1260 - 360 = 900 \]

### Final Answer
The number of ways in which the letters of the word "ARRANGE" can be arranged such that no two R's are together is **900**.

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