The number of words with or without meaning that can be formed using all the letter of the words "FARMER" such that both R do not appear together is . .

Answer

To solve the problem, we need to find the number of arrangements of the letters in the word "FARMER" such that the two 'R's do not appear together.

### Step-by-Step Solution:

1. **Count the total letters in "FARMER":**
The word "FARMER" consists of 6 letters: F, A, R, M, E, R. Here, we have 2 R's which are identical.

2. **Calculate the total arrangements of the letters:**
The total number of arrangements of the letters in "FARMER" can be calculated using the formula for permutations of multiset:
\[ \text{Total arrangements} = \frac{n!}{p_1! \cdot p_2! \cdots} \]
where \( n \) is the total number of letters and \( p_1, p_2, \ldots \) are the frequencies of the identical letters.
Here, we have:
\[ n = 6 \quad (\text{F, A, R, M, E, R}) \]
and the frequency of R is 2. Therefore, the total arrangements are:
\[ \text{Total arrangements} = \frac{6!}{2!} = \frac{720}{2} = 360 \]

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 unit or letter. Thus, we can think of the letters as: (RR), F, A, M, E. This gives us a total of 5 units to arrange:
\[ \text{Units} = (RR), F, A, M, E \]
The number of arrangements of these 5 units is:
\[ \text{Arrangements with RR together} = 5! = 120 \]

4. **Calculate 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 they 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:
\[ \text{Arrangements with R's not together} = 360 - 120 = 240 \]

### Final Answer:
The number of words that can be formed using all the letters of the word "FARMER" such that both R's do not appear together is **240**.