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

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

 

Option: 1

625


Option: 2

1316


Option: 3

4352


Option: 4

772


Answers (1)

best_answer

Given that the word is GENUINE.

The lexicographic order of the letters of the given word is E, E, G, I, N, N, 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 seven-letter word, then the remaining four letters can be arranged in 6! ways . On proceeding like this, 

\begin{aligned} & \text { E }(-----)=\frac{6 !}{2 !} \text { ways } \\ & \operatorname{GEE}(----)=\frac{4 !}{2 !} \text { ways } \\ & \operatorname{GEI}(----)=\frac{4 !}{2 !} \text { ways } \\ & \operatorname{GENE}(---)=3 \text { ! ways } \\ & \operatorname{GENI}(---)=3 ! \text { ways } \\ \end{aligned}

\begin{aligned} & \operatorname{GENN}(---)=3 \text { ! ways } \\ & \operatorname{GENUE}(--)=2 \text { ! ways } \\ & \operatorname{GENUIE}(-)=1 \text { ! ways } \\ & \operatorname{GENUINE~}=0 \text { ! ways } \end{aligned}

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


\begin{aligned} & =\frac{6 !}{2 !}+\frac{6 !}{2 !}+\frac{4 !}{2 !}+\frac{4 !}{2 !}+3 !+3 !+3 !+2 !+1 !+0 ! \\ & =(6 \times 5 \times 4 \times 3)+(6 \times 5 \times 4 \times 3)+(4 \times 3)+(4 \times 3)+(3 \times 2 \times 1)+(3 \times 2 \times 1)+(3 \times 2 \times 1)+(2 \times 1)+1+1 \\ & =360+360+12+12+6+6+6+6+2+1+1 \\ & =772 \end{aligned}

Therefore, the rank of the word GENUINE is 772

 

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vinayak

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