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

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

 

Option: 1

625


Option: 2

309


Option: 3

506


Option: 4

964


Answers (1)

best_answer

Given that the word is AWESOME. 

The lexicographic order of the letters of the given word is A, E, E, M, O, S, W. The words that begin with A will come first in the lexicographic order.

If the letter A is in the first place of the seven-letter word, then the remaining four letters can be arranged in 6! ways. On proceeding like this, 

\begin{aligned} & \mathrm{AE}(----)=5 ! \text { ways } \\ & \mathrm{AE}(----)=5 ! \text { ways } \\ & \mathrm{AM}(-----)=\frac{5 !}{2 !} \text { ways } \\ & \mathrm{AO}(-----)=\frac{5 !}{2 !} \text { ways } \\ & \mathrm{AS}(-----)=\frac{5 !}{2 !} \text { ways } \\ & \mathrm{AWEE}(----)=4 ! \text { ways } \\ & \mathrm{AWEM}(----)=4 ! \text { ways } \end{aligned}

\begin{aligned} & \operatorname{AWEO}(----)=4 \text { ! ways } \\ & \operatorname{AWESE}(---)=3 ! \text { ways } \\ & \operatorname{AWESM}(---)=3 ! \text { ways } \\ & \operatorname{AWESOE}(-)=1 ! \text { ways } \\ & \operatorname{AWESOME}=0 \text { ! ways } \end{aligned}

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

\begin{aligned} & =5 !+5 !+\frac{5 !}{2 !}+\frac{5 !}{2 !}+\frac{5 !}{2 !}+4 !+4 !+4 !+3 !+3 !+1 !+0 ! \\ & =(5 \times 4 \times 3 \times 2 \times 1)+(5 \times 4 \times 3 \times 2 \times 1)+(5 \times 4 \times 3)+(5 \times 4 \times 3)+(5 \times 4 \times 3) \\ & +(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) \\ & +(3 \times 2 \times 1)+(3 \times 2 \times 1)+1+1 \\ & =120+120+60+60+60+24+24+24+6+6+1+1 \\ & =506 \end{aligned}

Therefore, the rank of the word AWESOME is 506

 

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Rishi

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