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  • If the letters of the word SMILE are arranged in all possible ways and the words are arranged in a dictionary, then find the rank of the word SMILE.Option: 1</st

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

Option: 1

92


Option: 2

118


Option: 3

93


Option: 4

95


Answers (1)

best_answer

Given that the word is SMILE. 

The lexicographic order of the letters of the given word is E, I, L, M, S. The words that begin with E will come first in the lexicographic order.

If the letter E is in the first place of the five-letter word, then the remaining four letters can be arranged in 4! 

 ways. On proceeding like this, 

E(----)=4 ! \text { ways }

\begin{aligned} & \mathrm{I}(----)=4 ! \text { ways } \\ & \mathrm{L}(----)=4 ! \text { ways } \\ & \mathrm{M}(----)=4 ! \text { ways } \\ & \mathrm{SE}(---)=3 ! \text { ways } \\ & \mathrm{SI}(---)=3 ! \text { ways } \\ & \mathrm{SL}(---)=3 ! \text { ways } \\ & \mathrm{SME}(--)=2 ! \text { ways } \\ & \mathrm{SMIE}(-)=1 ! \text { ways } \end{aligned}

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

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

\begin{aligned} & =4 !+4 !+4 !+4 !+3 !+3 !+3 !+2 !+1 !+0 ! \\ & =(4 \times 3 \times 2 \times 1)+(4 \times 3 \times 2 \times 1)+(4 \times 3 \times 2 \times 1)+(4 \times 3 \times 2 \times 1)+ \\ & (3 \times 2 \times 1)+(3 \times 2 \times 1)+(3 \times 2 \times 1)+(2 \times 1)+1+1 \\ & =24+24+24+24+6+6+6+2+1+1 \\ & =118 \end{aligned}

Therefore, the rank of the word SMILE is 118

 

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