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

Option: 1

625


Option: 2

309


Option: 3

2018


Option: 4

964


Answers (1)

best_answer

Given that the word is ADORABLE. 

The lexicographic order of the letters of the given word is A, A, B, D, E, L, O, R. 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 6! ways. On proceeding like this, 

\begin{aligned} & \mathrm{AA}(-----)=6 ! \text { ways } \\ & \operatorname{AB}(------)=6 ! \text { ways } \\ & \operatorname{ADA}(-----)=5 ! \text { ways } \\ & \operatorname{ADB}(-----)=5 ! \text { ways } \\ & \operatorname{ADE}(-----)=5 ! \text { ways } \\ & \operatorname{ADL}(-----)=5 ! \text { ways } \\ & \operatorname{ADOA}(----)=4 ! \text { ways } \end{aligned}

\begin{aligned} & \operatorname{ADOB}(----)=4 ! \text { ways } \\ & \operatorname{ADOE}(----)=4 ! \text { ways } \\ & \operatorname{ADOL}(----)=4 ! \text { ways } \\ & \operatorname{ADORABE}(-)=1 ! \text { ways } \\ & \operatorname{ADORABLE}=0 ! \text { ways } \end{aligned}

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

\begin{aligned} & =6 !+6 !+5 !+5 !+5 !+5 !+4 !+4 !+4 !+4 !+1 !+0 ! \\ & =(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) \\ & +(5 \times 4 \times 3 \times 2 \times 1)+(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)+(4 \times 3 \times 2 \times 1)+(4 \times 3 \times 2 \times 1)+(4 \times 3 \times 2 \times 1)+1+1 \\ & =720+720+120+120+120+120+24+24+24+24+1+1 \\ & =2018 \end{aligned}

Therefore, the rank of the word ADORABLE is 2018

 

 

Posted by

Gautam harsolia

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