The number of arrangements of the letters of the word BANANA is which the two ‘N’s do not appear adjacently is

Options

A. 40
B. 60
C. 80
D. 100

Answer

To find the number of arrangements of the letters in the word "BANANA" such that the two 'N's do not appear adjacently, we can follow these steps:

### Step 1: Calculate the total arrangements without restrictions
The word "BANANA" consists of 6 letters: B, A, N, A, N, A. The letters include:
- B: 1
- A: 3
- N: 2

The formula for permutations of a multiset is given by:

\[ \text{Total arrangements} = \frac{n!}{n_1! \times n_2! \times n_3!} \]

Where:
- \( n \) is the total number of letters,
- \( n_1, n_2, n_3 \) are the frequencies of each distinct letter.

Thus, we have:

\[ \text{Total arrangements} = \frac{6!}{3! \times 2!} \]

Calculating this gives:

\[ 6! = 720, \quad 3! = 6, \quad 2! = 2 \]

So:

\[ \text{Total arrangements} = \frac{720}{6 \times 2} = \frac{720}{12} = 60 \]

### Step 2: 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. Thus, we can think of the arrangement as:

- NN (as one unit)
- B
- A
- A
- A

This gives us 5 units: NN, B, A, A, A.

Now, we calculate the arrangements of these 5 units:

\[ \text{Arrangements with NN together} = \frac{5!}{3!} \]

Calculating this gives:

\[ 5! = 120, \quad 3! = 6 \]

So:

\[ \text{Arrangements with NN together} = \frac{120}{6} = 20 \]

### Step 3: Calculate arrangements where the two 'N's do not appear together
To find the arrangements where the two 'N's do not appear together, we subtract the number of arrangements where they are together from the total arrangements:

\[ \text{Arrangements where N's are not together} = \text{Total arrangements} - \text{Arrangements with NN together} \]

Substituting the values we calculated:

\[ \text{Arrangements where N's are not together} = 60 - 20 = 40 \]

### Final Answer
Thus, the number of arrangements of the letters of the word "BANANA" in which the two 'N's do not appear adjacently is **40**.

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