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If the letters of the word CLEAN are arranged in all possible ways and the words are arranged in a dictionary, then find the number of words in the dictionary before the word CLEAN.

Option: 1

63


Option: 2

38


Option: 3

84


Option: 4

80


Answers (1)

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Given that the word is CLEAN. 

The lexicographic order of the letters of the given word is A, C, E, L, N. 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, 

\begin{aligned} & \mathrm{A}(----)=4 ! \text { ways } \\ & \mathrm{CA}(---)=3 ! \text { ways } \\ & \mathrm{CE}(---)=3 ! \text { ways } \\ & \mathrm{CLA}(--)=2 ! \text { ways } \\ & \mathrm{CLEAN}=0 ! \text { ways } \end{aligned}

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

\begin{aligned} & =4 !+3 !+3 !+2 !+0 ! \\ & =(4 \times 3 \times 2 \times 1)+(3 \times 2 \times 1)+(3 \times 2 \times 1)+(2 \times 1)+1 \\ & =24+6+6+2+1 \\ & =39 \end{aligned}

Therefore, the rank of the word CLEAN is 39

Thus, the number of words in a dictionary before the word CLEAN is  39-1=38.

 

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Riya

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