Question

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

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

Answer 1

The correct option is A

$40$

Explanation for correct answer :

Finding the number of arrangements:

There is a total of $6$ letters in the word ‘BANANA’ out of which N repeats $2$ times and A repeats $3$ times.

So, the total number of arrangements of BANANA is

$$\begin{gathered} \frac{\left(6!\right)}{\left(2!\right)\left(3!\right)} \\ =\frac{6\times 5\times 4\times 3\times 2\times 1}{2\times 1\times 3\times 2\times 1}=60 \end{gathered}$$

Now, taking two N's as one letter we'll get the total number of words in which Two N's ARE always together.

So, the number of arrangements in which the two N’s ARE together is B A A A NN

$$\begin{gathered} \frac{\left(5!\right)}{\left(3!\right)} \\ =\frac{5\times 4\times 3\times 2\times 1}{3\times 2\times 1}=\frac{120}{6}=20 \end{gathered}$$

The number of arrangements in which the two N’s do not appear adjacently is

$$\begin{gathered} =\text{Total number of arrangements}-\text{Number of arrangements where two N’s appear together} \\ =60-20 \\ =40 \end{gathered}$$

i.e. the number of arrangements of the letters of the word BANANA in which the two N’s do not appear adjacently is $40$.

Therefore, the correct answer is option (A).