Question

If the letters of the word 'CYCLE' all arranges to form a dictionary, then find: i) rank of 'CYCLE' ii) 35th word.

• A. i) 20, ii) EYCLC
• B. i) 19, ii) EYCLC
• C. i) 20, ii) CYCEL
• D. i) 19, ii) CYCEL

Answer 1

Answer: (A)

i) 20, ii) EYCLC

Answer 2

To find the rank of 'CYCLE': The letters are C, C, E, L, Y. Sorted unique letters: C, E, L, Y.

1. Words starting with C: Remaining (C, E, L, Y). Permutations = $4! = 24$.
2. Words starting with CY: Remaining (C, E, L).
   • CYC...: Remaining (E, L).
     • CYCE...: Remaining (L). Permutations = $1! = 1$ (CYCEL).
   • Words starting with C and second letter before Y (C, E, L):
     • CC...: Remaining (E, L, Y). Permutations = $3! = 6$.
     • CE...: Remaining (C, L, Y). Permutations = $3! = 6$.
     • CL...: Remaining (C, Y, E). Permutations = $3! = 6$. Total words before CYCLE = $6 + 6 + 6 + 1 = 19$. Rank of CYCLE = $19 + 1 = 20$.

To find the 35th word: Total permutations = $5! / 2! = 60$.

1. Words starting with C: $4! = 24$ (Ranks 1-24).
2. Words starting with E: Remaining (C, C, L, Y). Permutations = $4! / 2! = 12$ (Ranks 25-36). The 35th word is in the 'E' block. We need the $(35 - 24) = 11$th word in this block. Remaining letters for E-block: C, C, L, Y. Sorted: C, C, L, Y.
   • EC...: Remaining (C, L, Y). Permutations = $3! = 6$ (Ranks 1-6 in E-block).
   • EL...: Remaining (C, C, Y). Permutations = $3! / 2! = 3$ (Ranks 7-9 in E-block).
   • EY...: Remaining (C, C, L). Permutations = $3! / 2! = 3$ (Ranks 10-12 in E-block). We need the 11th word, which is the 2nd word in the 'EY' block. Words starting with EY, using C, C, L: 1st (10th in E-block): EYCCL 2nd (11th in E-block): EYCLC 3rd (12th in E-block): EYLCC The 35th word is EYCLC.