The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order s in an English dictionary. The number of words that appear before the word COCHIN is a.`360` b. `192` c. `96` d. `48`

Options

A. 360
B. 192
C. 96
D. 48

Explanation

To find the number of words that appear before the word "COCHIN" when all permutations of its letters are arranged in alphabetical order, we can follow these steps:

### Step 1: Identify the letters and their arrangement
The letters of the word "COCHIN" are C, O, C, H, I, N. We have a total of 6 letters where 'C' appears twice.

### Step 2: Arrange the letters in alphabetical order
The alphabetical order of the letters is:
- C, C, H, I, N, O

### Step 3: Fix the first letter
Since the first letter of "COCHIN" is 'C', we cannot have any letter before 'C'. Therefore, we move to the next letter.

### Step 4: Fix the second letter
The second letter in "COCHIN" is 'O'. We need to count how many permutations can be formed with letters that come before 'O' (i.e., 'C', 'H', 'I', 'N').

### Step 5: Count permutations with 'C' as the second letter
If we fix 'C' as the first letter and 'C' as the second letter, the remaining letters are H, I, N, O. The number of permutations of these 4 letters is given by:
\[ \frac{4!}{1!} = 24 \text{ (since 'C' is repeated)} \]

### Step 6: Count permutations with 'H' as the second letter
If we fix 'C' as the first letter and 'H' as the second letter, the remaining letters are C, I, N, O. The number of permutations of these 4 letters is:
\[ \frac{4!}{1!} = 24 \]

### Step 7: Count permutations with 'I' as the second letter
If we fix 'C' as the first letter and 'I' as the second letter, the remaining letters are C, H, N, O. The number of permutations of these 4 letters is:
\[ \frac{4!}{1!} = 24 \]

### Step 8: Count permutations with 'N' as the second letter
If we fix 'C' as the first letter and 'N' as the second letter, the remaining letters are C, H, I, O. The number of permutations of these 4 letters is:
\[ \frac{4!}{1!} = 24 \]

### Step 9: Total permutations before "COCHIN"
Now we can sum all the permutations counted:
- With 'C' as the second letter: 24
- With 'H' as the second letter: 24
- With 'I' as the second letter: 24
- With 'N' as the second letter: 24

Total = 24 + 24 + 24 + 24 = 96

### Conclusion
The total number of words that appear before "COCHIN" is **96**.