To solve the problem of arranging the letters of the word "BANANA" such that no two 'N's are together, we can follow these steps:
### Step 1: Identify the letters and their frequencies
The word "BANANA" consists of the following letters:
- B: 1 time
- A: 3 times
- N: 2 times
### Step 2: Calculate the total arrangements without any restrictions
To find the total arrangements of the letters in "BANANA", we use the formula for permutations of multiset:
\[
\text{Total arrangements} = \frac{n!}{p_1! \times p_2! \times p_3!}
\]
Where:
- \(n\) is the total number of letters,
- \(p_1, p_2, p_3\) are the frequencies of the distinct letters.
Here, \(n = 6\) (total letters), \(p_1 = 1\) (for B), \(p_2 = 3\) (for A), and \(p_3 = 2\) (for N).
So, we calculate:
\[
\text{Total arrangements} = \frac{6!}{1! \times 3! \times 2!}
\]
Calculating factorials:
\[
6! = 720, \quad 3! = 6, \quad 2! = 2
\]
Now substituting these values:
\[
\text{Total arrangements} = \frac{720}{1 \times 6 \times 2} = \frac{720}{12} = 60
\]
### Step 3: Calculate arrangements where the two 'N's are together
To find the arrangements where the two 'N's are together, we can treat the two 'N's as a single unit or block. Thus, we have the following letters to arrange:
- NN (as one block)
- B: 1 time
- A: 3 times
Now we have a total of 5 units to arrange: NN, B, A, A, A.
Using the same formula for permutations of multiset:
\[
\text{Arrangements with NN together} = \frac{5!}{1! \times 3!}
\]
Calculating factorials:
\[
5! = 120, \quad 3! = 6
\]
Now substituting these values:
\[
\text{Arrangements with NN together} = \frac{120}{1 \times 6} = \frac{120}{6} = 20
\]
### Step 4: Calculate arrangements where the two 'N's are not together
To find the arrangements where the two 'N's are not together, we subtract the arrangements where they are together from the total arrangements:
\[
\text{Arrangements with N's not together} = \text{Total arrangements} - \text{Arrangements with NN together}
\]
Substituting the values we calculated:
\[
\text{Arrangements with N's not together} = 60 - 20 = 40
\]
### Final Answer
The number of ways to arrange the letters of the word "BANANA" such that no two 'N's appear together is **40**.
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