In how many ways can the letters of the word ARRANGE be arranged so that the two R's are never together.

Answer

To solve the problem of arranging the letters of the word "ARRANGE" such that the two R's are never together, we can follow these steps:

### Step 1: Count 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 each appear 1 time

The formula for the total arrangements of letters, considering the repetitions, is given by:

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

Where \( n \) is the total number of letters, and \( p_1, p_2, \ldots \) are the frequencies of the repeating letters.

So, we have:

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

Calculating this:

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

### Step 2: Count the arrangements where the two R's are together
To count 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 (as one unit)
- A
- A
- N
- G
- E

This gives us a total of 6 units to arrange (RR, A, A, N, G, E).

Using the same formula for arrangements:

\[ \text{Arrangements with RR together} = \frac{6!}{2!} \]

Calculating this:

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

### Step 3: Calculate the arrangements where the two R's are never together
To find the arrangements where the two R's are never together, we subtract the arrangements where the R's are together from the total arrangements:

\[ \text{Arrangements with R's never together} = \text{Total arrangements} - \text{Arrangements with RR together} \]

Calculating this:

\[ \text{Arrangements with R's never together} = 1260 - 360 = 900 \]

### Final Answer
Thus, the number of ways to arrange the letters of the word "ARRANGE" such that the two R's are never together is **900**.

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