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

 

Option: 1

92


Option: 2

110


Option: 3

93


Option: 4

95


Answers (1)

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Given that the word is SPARK. 

The lexicographic order of the letters of the given word is A, K, P, R, S. The words that begin with A will come first in the lexicographic order.

If the letter A 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, 

\mathrm{A}(----)=4 !\: ways\\ $$ K(----)=4 !\: ways\\ $$ P(----)=4 !\: ways

\begin{aligned} & \mathrm{R}(----)=4 ! \text { ways } \\ & \mathrm{SA}(---)=3 ! \text { ways } \\ & \mathrm{SK}(---)=3 ! \text { ways } \\ & \operatorname{SPAK}(-)=1 ! \text { ways } \\ & \mathrm{SPARK}=0 \text { ! ways } \end{aligned}

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


\begin{aligned} & =4 !+4 !+4 !+4 !+3 !+3 !+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)+1+1 \\ & =24+24+24+24+6+6+1+1 \\ & =110 \end{aligned}

Therefore, the rank of the word SPARK is 110

 

 

Posted by

Ajit Kumar Dubey

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