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  • If the letters of the word JUBILANT are arranged in all possible ways and the words are arranged in a dictionary, then find the rank of the word JUBILANT.  <div class='qna-option

If the letters of the word JUBILANT are arranged in all possible ways and the words are arranged in a dictionary, then find the rank of the word JUBILANT.

 

Option: 1

19591


Option: 2

6971


Option: 3

3879


Option: 4

23879


Answers (1)

Given that the word is JUBILANT. 

The lexicographic order of the letters of the given word is A, B, I, J, L, N, T, U. The words that begin with A will come first in the lexicographic order.

If the letter A is in the first place of the eight-letter word, then the remaining four letters can be arranged in 7! ways. On proceeding like this, 

 \begin{aligned} & \mathrm{A}(------)=7 ! \text { ways } \\ & \mathrm{B}(------)=7 ! \text { ways } \\ & \mathrm{I}(-----)=7 ! \text { ways } \\ & \mathrm{JA}(------)=6 ! \text { ways } \\ & \mathrm{JB}(-----)=6 ! \text { ways } \\ & \mathrm{J}(------)=6 ! \text { ways } \\ & \mathrm{JL}(------)=6 ! \text { ways } \end{aligned}

\begin{aligned} & \mathrm{JN}(------)=6 ! \text { ways } \\ & \mathrm{JT}(------)=6 ! \text { ways } \\ & \mathrm{JUA}(-----)=5 ! \text { ways } \\ & \text { JUBA }(----)=4 ! \text { ways } \\ & \text { JUBIA }(---)=3 ! \text { ways } \\ & \text { JUBILANT }=0 ! \text { ways } \end{aligned}

So, the rank of the word JUBILANT is given by,


\begin{aligned} &= 7 !+7 !+7 !+6 !+6 !+6 !+6 !+6 !+6 !+5 !+4 !+3 !+0 ! \\ &=(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)+(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)+(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1) \\ &+(6 \times 5 \times 4 \times 3 \times 2 \times 1)+(6 \times 5 \times 4 \times 3 \times 2 \times 1)+(6 \times 5 \times 4 \times 3 \times 2 \times 1) \\ &+(6 \times 5 \times 4 \times 3 \times 2 \times 1)+(6 \times 5 \times 4 \times 3 \times 2 \times 1)+(6 \times 5 \times 4 \times 3 \times 2 \times 1)+ \\ &+(5 \times 4 \times 3 \times 2 \times 1)+(4 \times 3 \times 2 \times 1)+(3 \times 2 \times 1)+1 \\ &= 5040+5040+5040+720+720+720+720+720+720+120+24+6+1 \\ &= 19591 \end{aligned}

Therefore, the rank of the word JUBILANT is  19591

Posted by

Sumit Saini

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