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

Hint: First we find the total number of arrangements of the letters of the word 'BANANA'.
Then take two of the N's together i.e., take ‘NN’ as one block, and find the number of such permutations where the N’s appear together or adjacently. Now, note that the required number of permutations of the letters of the word 'BANANA' in which the two N’s do not appear adjacently is given by
$$=$$Total number of arrangements $$-$$Number of arrangements where two N’s appear together

Complete step by step Answer:

There is a total of 6 letters in the word ‘BANANA’ out of which N repeats 2 times and A repeats 3 times.
The total number of arrangements of the letters of the word 'BANANA' is $${\displaystyle \frac{6!}{3!2!}}={\displaystyle \frac{720}{6\times 2}}={\displaystyle \frac{720}{12}}=60$$ [divide by $$\left(3!2!\right)$$due to the repetitions of A and N]
Now we write two of the N's together i.e. take ‘NN’ as one block,
B A A A NN
So now basically we have 5 letters in total.
Therefore, Number of such permutations where the two N’s appear together or adjacently, is ${\displaystyle \frac{5!}{3!}}={\displaystyle \frac{120}{6}}=20$ [considering the repetitions of 3A′s]
Hence, the required number of permutations of the letters of the word 'BANANA' in which the two N’s do not appear adjacently is given by
$$=$$Total number of arrangements $$-$$Number of arrangements where two N’s appear together
$$=60-20$$
$$=40$$
The number of arrangements of the letters of the word 'BANANA' in which the two N’s do not appear adjacently is 40.

Note: A permutation is an act of arranging the objects or numbers in order while Combinations are the way of selecting the objects or numbers from a group of objects or collection, in such a way that the order of the objects does not matter.

The formula for permutations is given by: $${}^n{P}_r={\displaystyle \frac{n!}{\left(n-r\right)!}}$$
The formula for combinations is given by: $${}^n{C}_r={\displaystyle \frac{n!}{r!\left(n-r\right)!}}$$
In the case of repetitions while arranging:
Since in the word BANANA, there are 3 A’s and 2 N’s, therefore while calculating the number of arrangements one must consider the repetitions of the letters. Note that if there are n things to be arranged in a row, among which things are of one kind, b things are of another kind, and c things are of another, then the total number of arrangements is given by ${\displaystyle \frac{n!}{a!b!c!}}$.