To find the rank of the word "ARRANGE" in the dictionary, we will follow these steps:
### Step 1: Arrange the letters in alphabetical order
The letters of the word "ARRANGE" are A, A, E, G, N, R, R. When arranged in alphabetical order, they are:
- A, A, E, G, N, R, R
### Step 2: Count the total arrangements starting with letters before 'A'
Since the first letter of "ARRANGE" is 'A', we will consider the arrangements that start with letters that come before 'A'. However, there are no letters before 'A' in this case, so we move to the next step.
### Step 3: Count arrangements starting with 'A'
Next, we will count the arrangements that start with 'A'. After fixing one 'A', we have the letters A, E, G, N, R, R left. The total arrangements of these letters can be calculated using the formula for permutations of multiset:
\[
\text{Total arrangements} = \frac{5!}{2!}
\]
Where 5! is the factorial of the number of letters left (5 letters) and 2! accounts for the two 'R's.
Calculating this gives:
\[
\frac{5!}{2!} = \frac{120}{2} = 60
\]
### Step 4: Count arrangements starting with 'AA'
Next, we will consider arrangements that start with 'AA'. After fixing 'AA', we have E, G, N, R, R left. The total arrangements of these letters is:
\[
\text{Total arrangements} = \frac{5!}{2!}
\]
Calculating this again gives:
\[
\frac{5!}{2!} = 60
\]
### Step 5: Count arrangements starting with 'AE'
Next, we consider arrangements starting with 'AE'. After fixing 'AE', we have A, G, N, R, R left. The total arrangements of these letters is:
\[
\text{Total arrangements} = \frac{5!}{2!}
\]
Calculating this gives:
\[
\frac{5!}{2!} = 60
\]
### Step 6: Count arrangements starting with 'AG'
Next, we consider arrangements starting with 'AG'. After fixing 'AG', we have A, E, N, R, R left. The total arrangements of these letters is:
\[
\text{Total arrangements} = \frac{5!}{2!}
\]
Calculating this gives:
\[
\frac{5!}{2!} = 60
\]
### Step 7: Count arrangements starting with 'AN'
Next, we consider arrangements starting with 'AN'. After fixing 'AN', we have A, E, G, R, R left. The total arrangements of these letters is:
\[
\text{Total arrangements} = \frac{5!}{2!}
\]
Calculating this gives:
\[
\frac{5!}{2!} = 60
\]
### Step 8: Count arrangements starting with 'AR'
Now we will consider arrangements starting with 'AR'. After fixing 'AR', we have A, E, G, N, R left. The total arrangements of these letters is:
\[
\text{Total arrangements} = 4!
\]
Calculating this gives:
\[
4! = 24
\]
### Step 9: Count arrangements starting with 'ARA'
Now we will consider arrangements starting with 'ARA'. After fixing 'ARA', we have E, G, N, R left. The total arrangements of these letters is:
\[
\text{Total arrangements} = 4!
\]
Calculating this gives:
\[
4! = 24
\]
### Step 10: Count arrangements starting with 'ARE'
Now we will consider arrangements starting with 'ARE'. After fixing 'ARE', we have A, G, N, R left. The total arrangements of these letters is:
\[
\text{Total arrangements} = 4!
\]
Calculating this gives:
\[
4! = 24
\]
### Step 11: Count arrangements starting with 'ARG'
Now we will consider arrangements starting with 'ARG'. After fixing 'ARG', we have A, E, N, R left. The total arrangements of these letters is:
\[
\text{Total arrangements} = 4!
\]
Calculating this gives:
\[
4! = 24
\]
### Step 12: Count arrangements starting with 'ARN'
Now we will consider arrangements starting with 'ARN'. After fixing 'ARN', we have A, E, G, R left. The total arrangements of these letters is:
\[
\text{Total arrangements} = 4!
\]
Calculating this gives:
\[
4! = 24
\]
### Step 13: Count arrangements starting with 'ARR'
Finally, we will consider arrangements starting with 'ARR'. After fixing 'ARR', we have A, E, G, N left. The total arrangements of these letters is:
\[
\text{Total arrangements} = 3!
\]
Calculating this gives:
\[
3! = 6
\]
### Step 14: Calculate the total rank
Now we sum all the arrangements calculated:
- From 'A': 60
- From 'AA': 60
- From 'AE': 60
- From 'AG': 60
- From 'AN': 60
- From 'AR': 24
- From 'ARA': 24
- From 'ARE': 24
- From 'ARG': 24
- From 'ARN': 24
- From 'ARR': 6
Adding these gives:
\[
60 + 60 + 60 + 60 + 60 + 24 + 24 + 24 + 24 + 24 + 6 = 342
\]
Thus, the rank of the word "ARRANGE" in the dictionary is **343** (including the word itself).
