To find the rank of the word 'NAAGL' when all permutations of the letters of the word 'LAGAN' are arranged in dictionary order, we can follow these steps:
### Step 1: Identify the letters and their frequency
The letters in 'LAGAN' are L, A, G, A, N.
- A appears 2 times
- L, G, N each appear 1 time
### Step 2: Calculate total permutations
The total number of permutations of the letters in 'LAGAN' can be calculated using the formula for permutations of multiset:
\[
\text{Total permutations} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!}
\]
Where \( n \) is the total number of letters, and \( p_1, p_2, \ldots, p_k \) are the frequencies of the distinct letters.
For 'LAGAN':
\[
\text{Total permutations} = \frac{5!}{2! \times 1! \times 1! \times 1!} = \frac{120}{2} = 60
\]
### Step 3: Count permutations starting with letters before 'N'
We need to count all the permutations that start with letters that come before 'N' in alphabetical order, which are 'A' and 'G'.
#### Count permutations starting with 'A':
- Remaining letters: L, A, G, N
- Total permutations = \(\frac{4!}{1! \times 2! \times 1!} = \frac{24}{2} = 12\)
#### Count permutations starting with 'G':
- Remaining letters: A, A, L, N
- Total permutations = \(\frac{4!}{2! \times 1! \times 1!} = \frac{24}{2} = 12\)
### Step 4: Count permutations starting with 'N'
Now, we focus on the permutations that start with 'N'. The next letter in 'NAAGL' is 'A'.
#### Count permutations starting with 'NA':
- Remaining letters: A, G, L
- Total permutations = \(3! = 6\)
### Step 5: Count permutations starting with 'NAA'
- Remaining letters: G, L
- Total permutations = \(2! = 2\)
### Step 6: Count permutations starting with 'NAAG'
- Remaining letter: L
- Total permutations = \(1! = 1\)
### Step 7: Calculate the rank of 'NAAGL'
Now, we can sum all the permutations counted:
- Permutations starting with 'A': 12
- Permutations starting with 'G': 12
- Permutations starting with 'N' and 'A': 6
- Permutations starting with 'NA' and 'A': 2
- Permutations starting with 'NAAG': 1
Adding these gives:
\[
12 + 12 + 6 + 2 + 1 = 33
\]
Since 'NAAGL' is the next permutation after these, its rank is:
\[
33 + 1 = 34
\]
Thus, the rank of 'NAAGL' is **34**.
