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

 

Option: 1

625


Option: 2

1316


Option: 3

380


Option: 4

964


Answers (1)

best_answer

Given that the word is GORGEOUS.

The lexicographic order of the letters of the given word is E, G, G, O, O, R, S, U. The words that begin with E will come first in the lexicographic order.

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

\begin{aligned} & \text { E }(------)=\frac{7 !}{4 !} \text { ways } \\ & \text { GE }(-----)=\frac{6 !}{2 !} \text { ways } \\ & \text { GG }(-----)=\frac{6 !}{2 !} \text { ways } \\ & \text { GOE }(----)=5 ! \text { ways } \\ & \text { GOG }(----)=5 ! \text { ways } \\ & \text { GOO }(-----)=5 ! \text { ways } \\ & \text { GORE }(----)=4 ! \text { ways } \\ & \text { GORGEOS }(-)=1 ! \text { ways } \\ & \end{aligned}

\text { GORGEOUS }=0 \text { ! ways }

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


\begin{aligned} & =\frac{7 !}{4 !}+\frac{6 !}{2 !}+\frac{6 !}{2 !}+5 !+5 !+5 !+4 !+1 !+0 ! \\ & =(7 \times 6 \times 5)+(6 \times 5 \times 4 \times 3)+(6 \times 5 \times 4 \times 3) \\ & +(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)+1+1 \\ & =210+360+360+120+120+120+24+1+1 \\ & =1316 \end{aligned}Therefore, the rank of the word GORGEOUS is 1316

 

 

Posted by

shivangi.bhatnagar

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