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 find the number of arrangements of the letters in the word "ARRANGE" where 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, G, and E appear 1 time each.

The formula for the total arrangements of letters when there are repetitions is given by:

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

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

So, we have:

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

Calculating this gives:

\[ = \frac{5040}{2 \times 2} = \frac{5040}{4} = 1260 \]

### 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. Thus, we can consider the letters as:

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

This gives us a total of 6 units to arrange (where (RR) is treated as one unit). The arrangement of these 6 units, considering the repetition of A, is given by:

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

Calculating this gives:

\[ = \frac{720}{2} = 360 \]

### 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 arrangements where the R's are together from the total arrangements:

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

Substituting the values we calculated:

\[ = 1260 - 360 = 900 \]

### Final Answer:
The number of words in which the two R's are not together is **900**.

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