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

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

 

Option: 1

567


Option: 2

642


Option: 3

876


Option: 4

369


Answers (1)

best_answer

Given that the word is SIMPLE. 

The lexicographic order of the letters of the given word is E, I, L, M, P, 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 six-letter word, then the remaining four letters can be arranged in 5! ways. On proceeding like this, 

$$ E(-----)=5 !\: ways

\begin{aligned} & \mathrm{I}(-----)=5 ! \text { ways } \\ & \mathrm{L}(-----)=5 ! \text { ways } \\ & \mathrm{M}(-----)=5 ! \text { ways } \\ & \mathrm{P}(-----)=5 \text { ! ways } \\ & \operatorname{SE}(----)=4 ! \text { ways } \\ \end{aligned}

$$ \begin{aligned} & \operatorname{SIE}(---)=3 ! \text { ways } \\ & \operatorname{SIL}(---)=3 ! \text { ways } \\ & \operatorname{SIME}(--)=2 ! \text { ways } \\ & \operatorname{SIML}(--)=2 ! \text { ways } \\ & \operatorname{SIMPE}(-)=1 ! \text { ways } \end{aligned} $$\:

\\SIMPLE = 0! \: ways

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

\begin{aligned} & =5 !+5 !+5 !+5 !+5 !+4 !+3 !+3 !+2 !+2 !+1 !+0 ! \\ & =(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)+(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)+(3 \times 2 \times 1)+(2 \times 1)+(2 \times 1)+1+1 \\ & =120+120+120+120+120+24+6+6+2+2+1+1 \\ & =642 \end{aligned}

Therefore, the rank of the word SIMPLE is 642

Posted by

HARSH KANKARIA

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