The number of different permutations of the letters of the word 'BANANA' is equal to :

Question

Accepted Answer

To determine the number of different permutations of the letters in the word "BANANA," we need to consider the repetition of letters. The word "BANANA" consists of 6 letters where 'A' appears 3 times, 'N' appears 2 times, and 'B' appears once.

The formula for calculating permutations of a multiset (a set with repeated elements) is given by:

$$ \text{Number of permutations} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!} $$

where $ n $ is the total number of letters, and $ p_1, p_2, \ldots, p_k $ are the frequencies of the repeated letters.

For "BANANA":
- Total number of letters, $ n = 6 $
- Frequency of 'A', $ p_1 = 3 $
- Frequency of 'N', $ p_2 = 2 $
- Frequency of 'B', $ p_3 = 1 $

Plugging these values into the formula, we get:

$$ \text{Number of permutations} = \frac{6!}{3! \times 2! \times 1!} $$

Calculating the factorials:

- $ 6! = 720 $
- $ 3! = 6 $
- $ 2! = 2 $
- $ 1! = 1 $

Now, substitute these into the formula:

$$ \text{Number of permutations} = \frac{720}{6 \times 2 \times 1} = \frac{720}{12} = 60 $$

Thus, the number of different permutations of the letters in "BANANA" is 60, which corresponds to Option 4.

**Explanation for Incorrect Options:**

1. **Option 1 (720):** This would be the number of permutations if all letters were unique. However, due to the repetition of letters, the actual number is reduced.

2. **Option 2 (120):** This number does not account for the correct reduction due to the repeated letters. It might be a result of incorrect calculation or misunderstanding of the formula.

3. **Option 3 (360):** This number is also incorrect as it does not properly account for the repetitions of 'A' and 'N'. It might be a result of partial consideration of repetitions.

Therefore, the correct answer, supported by the calculation, is Option 4, which is 60.