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Exercise 6.3 · Q18

Q.If the letters of the word ADINI are arranged as in dictionary, then what is the 46th word?

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List the words of ADINI in dictionary order block by block (fixing each letter position from left to right) until the 46th word is located.

[!FORMULA] Letters of ADINI, sorted alphabetically: A,D,I,I,NA, D, I, I, N (the letter II repeats twice). Total distinct arrangements =5!2!=60=\dfrac{5!}{2!}=60. For a fixed leading letter, the count of resulting words is (remaining letters)!repetition factorial of the remaining letters\dfrac{(\text{remaining letters})!}{\text{repetition factorial of the remaining letters}}.

  1. 1st letter = A: remaining letters D,I,I,ND,I,I,N give 4!2!=12\dfrac{4!}{2!}=12 words — positions 11–1212.
  2. 1st letter = D: remaining A,I,I,NA,I,I,N give 4!2!=12\dfrac{4!}{2!}=12 words — positions 1313–2424.
  3. 1st letter = I: one II is used, so remaining A,D,I,NA,D,I,N are all distinct, giving 4!=244!=24 words — positions 2525–4848. Since 25≤46≤4825\le46\le48, the 46th word starts with I; its local position within this block is 46−24=2246-24=22.
  4. 2nd letter (within I…), remaining pool {A,D,I,N}\{A,D,I,N\} taken in order: A→A\to words 2525–3030 (6 words), D→31D\to31–3636, I→37I\to37–4242, N→43N\to43–4848 (each block 3!=63!=6 words). Local position 2222 falls in the NN block (positions 4343–4848), sub-position 22−18=422-18=4. …
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