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

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

 

Option: 1

567


Option: 2

13951


Option: 3

3879


Option: 4

23879


Answers (1)

best_answer

Given that the word is FRIENDLY. 

The lexicographic order of the letters of the given word is D, E, F, I, L, N, R, Y. The words that begin with D will come first in the lexicographic order.

If the letter D 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{D}(-------)=7 ! \text { ways } \\ & \mathrm{E}(-------)=7 ! \text { ways } \\ & \mathrm{FD}(------)\: \: \: =6 ! \text { ways } \\ & \mathrm{FE}(------)\: \: \: =6 ! \text { ways } \\ & \mathrm{FI}(------)\: \: \: \: =6 ! \text { ways } \\ & \mathrm{FL}(------)\: \: \: =6 ! \text { ways } \\ & \mathrm{FRD}(-----)\:\: \: =5 ! \text { ways } \\ & \mathrm{FRE}(-----)\:\: \: =5 ! \text { ways } \\ & \mathrm{FRID}(----)\: \:\: \: \: \: =4 ! \text { ways } \\ & \mathrm{FRIEL}(---)\: \: \: \: \: \: =3 ! \text { ways } \end{aligned}

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

\begin{aligned} & =7 !+7 !+6 !+6 !+6 !+6 !+6 !+5 !+5 !+4 !+3 !+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)+(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)+(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 \\ & =5040+5040+720+720+720+720+720+120+120+24+6+1 \\ & =13951 \end{aligned}

Therefore, the rank of the word FRIENDLY is 13951

 

 

 

Posted by

jitender.kumar

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