If the letters of the word are arranged as inn a dictionary. M and n are the rank of the words BULBUL and NANNU respectively, then the value of m-4n is

Explanation

To solve the problem, we need to find the ranks of the words "BULBUL" and "NANNU" when arranged in alphabetical order, and then compute the value of \( m - 4n \).

### Step 1: Find the rank of the word "BULBUL"

1. **Arrange the letters in alphabetical order**: The letters of "BULBUL" are B, B, L, L, U, U. In alphabetical order, they are: B, B, L, L, U, U.

2. **Calculate the total arrangements starting with B**:
- Fix the first B. The remaining letters are B, L, L, U, U.
- The number of arrangements of these letters is given by:
\[ \frac{5!}{2! \cdot 2!} = \frac{120}{4} = 30 \]

3. **Calculate the arrangements starting with the second B**:
- Fix the second B. The remaining letters are B, L, L, U, U.
- The number of arrangements is the same as above:
\[ \frac{5!}{2! \cdot 2!} = 30 \]

4. **Calculate the arrangements starting with L**:
- Fix L. The remaining letters are B, B, L, U, U.
- The number of arrangements is:
\[ \frac{5!}{2! \cdot 2!} = 30 \]

5. **Calculate the arrangements starting with U**:
- Fix U. The remaining letters are B, B, L, L.
- The number of arrangements is:
\[ \frac{4!}{2! \cdot 2!} = 6 \]

6. **Sum the contributions**:
- The total number of words before "BULBUL" is:
\[ 30 + 30 + 30 + 6 = 96 \]
- Therefore, the rank \( m \) of "BULBUL" is \( 96 + 1 = 97 \).

### Step 2: Find the rank of the word "NANNU"

1. **Arrange the letters in alphabetical order**: The letters of "NANNU" are A, N, N, N, U. In alphabetical order, they are: A, N, N, N, U.

2. **Calculate the total arrangements starting with A**:
- Fix A. The remaining letters are N, N, N, U.
- The number of arrangements is:
\[ \frac{4!}{3!} = 4 \]

3. **Calculate the arrangements starting with N**:
- Fix N. The remaining letters are A, N, N, U.
- The number of arrangements is:
\[ \frac{4!}{2!} = 12 \]

4. **Calculate the arrangements starting with U**:
- Fix U. The remaining letters are A, N, N, N.
- The number of arrangements is:
\[ \frac{4!}{3!} = 4 \]

5. **Sum the contributions**:
- The total number of words before "NANNU" is:
\[ 4 + 12 = 16 \]
- Therefore, the rank \( n \) of "NANNU" is \( 16 + 1 = 17 \).

### Step 3: Calculate \( m - 4n \)

1. **Substituting the values**:
- We have \( m = 97 \) and \( n = 17 \).
- Calculate \( m - 4n \):
\[ m - 4n = 97 - 4 \times 17 = 97 - 68 = 29 \]

### Final Answer:
The value of \( m - 4n \) is \( 29 \).

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