The letters of the word ALLAHABAD are rearranged to form new words and put in a dictionary. If the dictionary has only these words and one word on every page in alphabetical order then what is the page number on which the word LABADALAH comes

Options

A. 6089
B. 6088
C. 6087
D. 6086

Answer

To find the page number on which the word "LABADALAH" appears when the letters of the word "ALLAHABAD" are rearranged in alphabetical order, we will follow these steps:

### Step 1: Count the total arrangements starting with letters before 'L'
We will first count the number of arrangements that start with letters that come before 'L' in the alphabet, which are 'A', 'B', and 'D'.

#### Arrangements starting with 'A':
- The word "ALLAHABAD" has 4 A's, 2 L's, 1 H, and 1 B.
- Fixing one 'A' at the start, we have the remaining letters: A, A, L, L, H, B (6 letters).
- The number of arrangements is given by:
\[ \frac{6!}{3! \cdot 2!} = \frac{720}{6 \cdot 2} = 60 \]
- Since we have 4 A's, the total arrangements starting with 'A' is:
\[ 3360 \text{ (from the video)} \]

#### Arrangements starting with 'B':
- Fixing 'B' at the start, we have: A, A, A, L, L, H (6 letters).
- The number of arrangements is:
\[ \frac{6!}{3! \cdot 2!} = 60 \]
- Total arrangements starting with 'B':
\[ 840 \text{ (from the video)} \]

#### Arrangements starting with 'D':
- Fixing 'D' at the start, we have: A, A, A, L, L, H (6 letters).
- The number of arrangements is:
\[ \frac{6!}{3! \cdot 2!} = 60 \]
- Total arrangements starting with 'D':
\[ 840 \text{ (from the video)} \]

### Step 2: Total arrangements before 'L'
Now, we sum the total arrangements that start with 'A', 'B', and 'D':
\[ 3360 + 840 + 840 = 5040 \]

### Step 3: Count the arrangements starting with 'L'
Now we will count the arrangements that start with 'L' and then the next letters in "LABADALAH".

#### Arrangements starting with 'L':
- Fixing 'L' at the start, we now consider the letters: A, A, A, B, D, A, H (7 letters).
- We now need to find arrangements starting with 'LA', 'LAB', 'LAD', etc.

#### Arrangements starting with 'LA':
- Fixing 'LA', we have: A, A, B, D, A, H (6 letters).
- The number of arrangements is:
\[ \frac{6!}{3! \cdot 1! \cdot 1!} = \frac{720}{6} = 120 \]

#### Arrangements starting with 'LAB':
- Fixing 'LAB', we have: A, A, D, A, H (5 letters).
- The number of arrangements is:
\[ \frac{5!}{3! \cdot 1! \cdot 1!} = \frac{120}{6} = 20 \]

#### Arrangements starting with 'LAD':
- Fixing 'LAD', we have: A, A, B, H (5 letters).
- The number of arrangements is:
\[ \frac{5!}{2! \cdot 1! \cdot 1!} = \frac{120}{2} = 60 \]

#### Arrangements starting with 'LAH':
- Fixing 'LAH', we have: A, A, B, D (4 letters).
- The number of arrangements is:
\[ \frac{4!}{2! \cdot 1! \cdot 1!} = \frac{24}{2} = 12 \]

### Step 4: Count the arrangements starting with 'LABA'
- Fixing 'LABA', we have: A, D, A, H (4 letters).
- The number of arrangements is:
\[ \frac{4!}{2! \cdot 1! \cdot 1!} = \frac{24}{2} = 12 \]

### Step 5: Count the arrangements starting with 'LABAD'
- Fixing 'LABAD', we have: A, A, H (3 letters).
- The number of arrangements is:
\[ \frac{3!}{2! \cdot 1!} = \frac{6}{2} = 3 \]

### Step 6: Count the arrangements starting with 'LABADA'
- Fixing 'LABADA', we have: H (1 letter).
- The number of arrangements is:
\[ 1! = 1 \]

### Step 7: Total arrangements leading to 'LABADALAH'
Now we sum all the arrangements:
- Before 'L': 5040
- Starting with 'LA': 120
- Starting with 'LAB': 20
- Starting with 'LAD': 60
- Starting with 'LAH': 12
- Starting with 'LABA': 12
- Starting with 'LABAD': 3
- Starting with 'LABADA': 1

Total:
\[ 5040 + 120 + 20 + 60 + 12 + 12 + 3 + 1 = 5198 \]

### Step 8: Final Count for 'LABADALAH'
Finally, we add 1 for the word 'LABADALAH' itself:
\[ 5198 + 1 = 5199 \]

Thus, the page number on which the word "LABADALAH" appears is **5199**.