The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is

$\text{Given: word } = \text{COCHIN}$

$\text{letters: } C, O, C, H, I, N \quad (C \text{ repeats twice})$

$\text{To find: number of words before COCHIN}$

$\text{Alphabetical order: } C < H < I < N < O \quad (\text{two } C\text{'s})$ $\text{Total permutations} = \dfrac{6!}{2!} = 360$ $\text{At each position, count available letters} < \text{COCHIN's letter}$ $\text{multiply by arrangements of the rest, then add}$ $\text{Pos 1 (letter } C): \ \text{no letter} < C \Rightarrow 0$ $\text{Pos 2 (letter } O): \ \text{smaller } C, H, I, N \Rightarrow 4 \text{ choices}$ $4 \times 4! = 4 \times 24 = 96$ $\text{Prefix } CO; \ \text{remaining } C, H, I, N$ $\text{Pos 3 (letter } C): \ 0$ $\text{Prefix } COC; \ \text{remaining } H, I, N$ $\text{Pos 4 (letter } H): \ 0$ $\text{Prefix } COCH; \ \text{remaining } I, N$ $\text{Pos 5 (letter } I): \ 0$ $\text{Prefix } COCHI; \ \text{remaining } N$ $\text{Pos 6 (letter } N): \ 0$ $0 + 96 + 0 + 0 + 0 + 0 = 96$ $\text{Rank of COCHIN} = 96 + 1 = 97$ $\text{Answer: } 96 \ \Rightarrow \ (3)$