Different words are being formed by arranging the letter of the word 'ARRANGE'
Q. The number of words in which the two R's are not together, is

Options

A. 1260
B. 960
C. 900
D. 600

Explanation

To solve the problem of finding the number of arrangements of the letters in the word "ARRANGE" such that the two R's are not 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 formula for the total arrangements of letters when there are repetitions is given by:

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

where \( n \) is the total number of letters, and \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeated letters.

Thus, the total arrangements of "ARRANGE" is:

\[ \text{Total arrangements} = \frac{7!}{2! \cdot 2! \cdot 1! \cdot 1! \cdot 1!} = \frac{5040}{4} = 1260 \]

### Step 2: Calculate the arrangements where the two R's are together.

To find the arrangements where the two R's are together, we can treat the two R's as a single unit or block. Therefore, we can consider the letters as:

- (RR), A, A, N, G, E

This gives us a total of 6 units to arrange, where A appears 2 times.

Using the same formula for arrangements:

\[ \text{Arrangements with RR together} = \frac{6!}{2! \cdot 1! \cdot 1! \cdot 1!} = \frac{720}{2} = 360 \]

### Step 3: Calculate the arrangements where the two R's are not together.

To find the arrangements where the two R's are not together, we subtract the number of arrangements where the R's are together from the total arrangements:

\[ \text{Arrangements with R's not together} = \text{Total arrangements} - \text{Arrangements with RR together} \]

Substituting the values we calculated:

\[ \text{Arrangements with R's not together} = 1260 - 360 = 900 \]

### Final Answer

Thus, the number of words in which the two R's are not together is **900**.

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