To find the rank of the word "CHENNAI" when all its letters are arranged in alphabetical order, we will follow these steps:
### Step 1: Arrange the letters in alphabetical order
The letters of "CHENNAI" are: A, C, E, H, I, N, N.
### Step 2: Calculate the total number of arrangements starting with letters before 'C'
1. **Starting with 'A':**
- Remaining letters: C, E, H, I, N, N (6 letters)
- Arrangements = \( \frac{6!}{2!} = \frac{720}{2} = 360 \)
2. **Starting with 'B':** (Not applicable as 'B' is not in "CHENNAI")
3. **Starting with 'C':** We now consider arrangements starting with 'C'.
### Step 3: Calculate arrangements starting with 'C' and letters before 'H'
1. **Starting with 'CA':**
- Remaining letters: E, H, I, N, N (5 letters)
- Arrangements = \( \frac{5!}{2!} = \frac{120}{2} = 60 \)
2. **Starting with 'CE':**
- Remaining letters: H, I, N, N (5 letters)
- Arrangements = \( \frac{5!}{2!} = \frac{120}{2} = 60 \)
3. **Starting with 'CH':** We now consider arrangements starting with 'CH'.
### Step 4: Calculate arrangements starting with 'CH' and letters before 'E'
1. **Starting with 'CHA':**
- Remaining letters: E, I, N, N (4 letters)
- Arrangements = \( \frac{4!}{2!} = \frac{24}{2} = 12 \)
2. **Starting with 'CHE':** We now consider arrangements starting with 'CHE'.
### Step 5: Calculate arrangements starting with 'CHE' and letters before 'N'
1. **Starting with 'CHEA':**
- Remaining letters: I, N, N (3 letters)
- Arrangements = \( \frac{3!}{2!} = \frac{6}{2} = 3 \)
2. **Starting with 'CHEI':**
- Remaining letters: A, N, N (3 letters)
- Arrangements = \( \frac{3!}{2!} = \frac{6}{2} = 3 \)
3. **Starting with 'CHEN':** Now we are at 'CHEN', which is the prefix of "CHENNAI".
### Step 6: Calculate arrangements starting with 'CHEN' and letters before 'N'
1. **Starting with 'CHENA':**
- Remaining letters: I, N (2 letters)
- Arrangements = \( 2! = 2 \)
2. **Starting with 'CHENI':**
- Remaining letters: A, N (2 letters)
- Arrangements = \( 2! = 2 \)
3. **Starting with 'CHENN':** Now we have 'CHENN', which is the prefix of "CHENNAI".
### Step 7: Calculate arrangements starting with 'CHENN' and letters before 'A'
1. **Starting with 'CHENNA':**
- Remaining letters: I (1 letter)
- Arrangements = \( 1! = 1 \)
### Step 8: Calculate the total rank of "CHENNAI"
Now we add all the arrangements calculated:
- Starting with 'A': 360
- Starting with 'CA': 60
- Starting with 'CE': 60
- Starting with 'CHA': 12
- Starting with 'CHEA': 3
- Starting with 'CHEI': 3
- Starting with 'CHENA': 2
- Starting with 'CHENI': 2
- Starting with 'CHENN': 1
Total = 360 + 60 + 60 + 12 + 3 + 3 + 2 + 2 + 1 = 503
### Step 9: Calculate the total arrangements of "CHENNAI"
Total arrangements of "CHENNAI" = \( \frac{7!}{2!} = \frac{5040}{2} = 2520 \)
### Step 10: Calculate the rank from the last
Rank from the last = Total arrangements - Rank from the top + 1
= \( 2520 - 503 + 1 = 2018 \)
Thus, the rank of the word "CHENNAI" from the last is **2018**.
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