In fall permutations of the letters of the 'LAGAN' are arranged as in dictionary, then what is the rank of 'NAAGL'?

Options

A. `48^(th)` Word
B. `49^(th)` Word
C. `50^(th)` Word
D. `51^(th)` Word

Answer

To find the rank of the word "NAAGL" when all permutations of the letters in "LAGAN" are arranged in dictionary order, we can follow these steps:

### Step 1: List the letters in alphabetical order
The letters in "LAGAN" are A, A, G, L, N. Arranged in alphabetical order, they are:
- A, A, G, L, N

### Step 2: Count permutations starting with letters before 'N'
We will count how many permutations can be formed with letters that come before 'N'.

1. **Starting with 'A':**
- The remaining letters are A, G, L, N.
- The number of permutations of these letters is calculated as:
\[ \frac{4!}{2!} = \frac{24}{2} = 12 \]
(We divide by 2! because 'A' is repeated.)

2. **Starting with 'G':**
- The remaining letters are A, A, L, N.
- The number of permutations of these letters is:
\[ \frac{4!}{2!} = \frac{24}{2} = 12 \]

3. **Starting with 'L':**
- The remaining letters are A, A, G, N.
- The number of permutations of these letters is:
\[ \frac{4!}{2!} = \frac{24}{2} = 12 \]

### Step 3: Total permutations before 'N'
Now we add the permutations starting with 'A', 'G', and 'L':
\[ 12 + 12 + 12 = 36 \]

### Step 4: Count permutations starting with 'N'
Now we consider the permutations starting with 'N'. The next letter in "NAAGL" is 'A'.

1. **Starting with 'NA':**
- The remaining letters are A, G, L.
- The number of permutations of these letters is:
\[ 3! = 6 \]

### Step 5: Count permutations starting with 'NAA'
Now we consider the permutations starting with 'NAA'. The next letter is 'G'.

1. **Starting with 'NAA':**
- The remaining letters are G, L.
- The number of permutations of these letters is:
\[ 2! = 2 \]

### Step 6: Count permutations starting with 'NAAG'
Finally, we consider the permutations starting with 'NAAG'.

1. **Starting with 'NAAG':**
- The remaining letter is L.
- There is only 1 permutation: "NAAGL".

### Step 7: Calculate the rank of 'NAAGL'
Now we can calculate the rank of "NAAGL":
- Total permutations before 'N': 36
- Permutations starting with 'NA': 6
- Permutations starting with 'NAA': 2
- The word "NAAGL" is the next one.

Thus, the rank of "NAAGL" is:
\[ 36 + 6 + 2 + 1 = 45 \]

### Final Answer
The rank of "NAAGL" is **45**.