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  • If the letters of the word GRACEFUL are arranged in all possible ways and the words are arranged in a dictionary, then find the rank of the word GRACEFUL.  <div class='qna-option

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

 

Option: 1

23761


Option: 2

6971


Option: 3

3879


Option: 4

23879


Answers (1)

best_answer

Given that the word is GRACEFUL. 

The lexicographic order of the letters of the given word is A, C, E, F, G, L, R, U. 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 7!  ways. On proceeding like this

\begin{aligned} & \mathrm{A}(-----)=7 ! \text { ways } \\ & \mathrm{C}(-----)=7 ! \text { ways } \\ & \mathrm{E}(-----)=7 ! \text { ways } \\ & \mathrm{F}(-----)=7 ! \text { ways } \\ & \text { GA }(-----)=6 ! \text { ways } \\ & \text { GC }(-----)=6 ! \text { ways } \\ & \text { GE }(-----)=6 ! \text { ways } \\ & \text { GF }(-----)=6 ! \text { ways } \\ & \text { GL }(-----)=6 ! \text { ways } \\ & \text { GRACEFUL } 0 \text { ! ways } \end{aligned}

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

\begin{aligned} & =7 !+7 !+7 !+7 !+6 !+6 !+6 !+6 !+6 !+0 ! \\ & =(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)+(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)+(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1) \\ & +(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)+(6 \times 5 \times 4 \times 3 \times 2 \times 1)+(6 \times 5 \times 4 \times 3 \times 2 \times 1) \\ & +(6 \times 5 \times 4 \times 3 \times 2 \times 1)+(6 \times 5 \times 4 \times 3 \times 2 \times 1)+(6 \times 5 \times 4 \times 3 \times 2 \times 1)+1 \\ & =5040+5040+5040+5040+720+720+720+720+720+1 \\ & =23761 \end{aligned}

Therefore, the rank of the word GRACEFUL is  23761

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