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

 

Option: 1

625


Option: 2

309


Option: 3

751


Option: 4

964


Answers (1)

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Given that the word is ELEGANT. 

The lexicographic order of the letters of the given word is A, E, E, G, L, N, T. 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{A}(-----)=\frac{6 !}{2 !} \text { ways } \\ & \text { EA }(----)=5 ! \text { ways } \\ & \text { EE }(----)=5 ! \text { ways } \\ & \text { EG }(----)=5 ! \text { ways } \\ & \text { ELA }(----)=4 ! \text { ways } \end{aligned}

\operatorname{ELEA}(---)-3 ! ways

ELEGANT $=0$ ! \; way

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

\begin{aligned} & =\frac{6 !}{2 !}+5 !+5 !+5 !+4 !+3 !+0 ! \\ & =(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)+(3 \times 2 \times 1)+1 \\ & =360+120+120+120+24+6+1 \\ & =751 \end{aligned}

Therefore, the rank of the word ELEGANT is 751

 

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Riya

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