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: Count the total letters and their frequencies
The word "ARRANGE" consists of 7 letters: A, R, R, A, N, G, E. The frequencies of the letters are:
- A: 2
- R: 2
- N: 1
- G: 1
- E: 1
### Step 2: Calculate the total arrangements without restrictions
The total number of arrangements of the letters in "ARRANGE" can be calculated using the formula for permutations of multiset:
\[
\text{Total arrangements} = \frac{7!}{2! \times 2!}
\]
Where \(7!\) is the factorial of the total number of letters, and \(2!\) accounts for the repetitions of A's and R's.
Calculating this gives:
\[
7! = 5040
\]
\[
2! = 2
\]
Thus,
\[
\text{Total arrangements} = \frac{5040}{2 \times 2} = \frac{5040}{4} = 1260
\]
### Step 3: Calculate 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 entity or block. Therefore, we can consider the block "RR" along with the other letters A, A, N, G, and E.
This gives us the letters: RR, A, A, N, G, E (6 letters in total).
The number of arrangements of these 6 letters is:
\[
\text{Arrangements with R's together} = \frac{6!}{2!}
\]
Where \(6!\) is the factorial of the total letters (including the block RR) and \(2!\) accounts for the repetitions of A's.
Calculating this gives:
\[
6! = 720
\]
Thus,
\[
\text{Arrangements with R's together} = \frac{720}{2} = 360
\]
### Step 4: Calculate arrangements where no two R's are together
To find the arrangements where no two R's are 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:
\[
\text{Arrangements with R's not together} = 1260 - 360 = 900
\]
### Final Answer
Thus, 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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