The difference between the different permutation of the word BANANA and rank of the word BANANA is

(A) 60

(B) 35

(C) 24

(D) None of these

To tackle the problem of finding the difference between the number of different permutations of the word "BANANA" and the rank of the word "BANANA," we first need to break down both components: the total permutations and the rank calculation.

Calculating Total Permutations

The word "BANANA" consists of 6 letters where 'A' appears 3 times, 'N' appears 2 times, and 'B' appears 1 time. The formula for calculating permutations of a multiset is given by:

P(n; n1, n2, ..., nk) = n! / (n1! * n2! * ... * nk!)

In our case:

• n = 6 (total letters)
• n1 = 3 (for A)
• n2 = 2 (for N)
• n3 = 1 (for B)

Plugging in the values, we get:

P(6; 3, 2, 1) = 6! / (3! * 2! * 1!)

Calculating the factorials:

• 6! = 720
• 3! = 6
• 2! = 2
• 1! = 1

Now substituting these values:

P(6; 3, 2, 1) = 720 / (6 * 2 * 1) = 720 / 12 = 60

Finding the Rank of the Word "BANANA"

The rank of a word among its permutations is determined by counting how many permutations would come before it in alphabetical order. To find the rank of "BANANA," we can follow these steps:

1. **List the letters in alphabetical order**: A, A, A, B, N, N.

2. **Fix the first letter and count permutations of the remaining letters**:

Starting with 'A':

• Fixing 'A', the remaining letters are A, A, B, N, N.
• Permutations = 5! / (3! * 2!) = 10.

Now, fixing 'B':

• Next, we fix 'B' and consider the letters A, A, A, N, N.
• Since 'B' is fixed, we need to count permutations of A, A, A, N, N.
• Permutations = 5! / (3! * 2!) = 10.

Next, we fix 'N':

• Fixing 'N', we have A, A, A, B, N.
• Permutations = 5! / (3! * 1! * 1!) = 20.

Now, we can sum these counts:

Rank of "BANANA" = 10 (for A) + 10 (for B) + 0 (for N) + 1 (itself) = 21

Calculating the Difference

Now that we have both values:

• Total permutations of "BANANA" = 60
• Rank of "BANANA" = 21

To find the difference:

Difference = Total Permutations - Rank = 60 - 21 = 39

Final Thoughts

Since the difference of 39 does not match any of the provided options (A) 60 (B) 35 (C) 24 (D) None of these, the correct answer is (D) None of these.