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  • If the letters of the word PROUD 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 PROUD. &nbsp

If the letters of the word PROUD 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 PROUD.

 

Option: 1

63


Option: 2

89


Option: 3

93


Option: 4

95


Answers (1)

best_answer

Given that the word is PROUD. 

The lexicographic order of the letters of the given word is D, O, P, R, U. The words that begin with D will come first in the lexicographic order.

If the letter D 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{D}(----)=4 ! \text { ways } \\ & \mathrm{O}(----)=4 ! \text { ways } \\ & \mathrm{PD}(---)=3 ! \text { ways } \\ & \mathrm{PO}(---)=3 ! \text { ways } \\ & \mathrm{PRD}(--)=2 ! \text { ways } \\ & \operatorname{PROD}(-)=1 ! \text { ways } \\ & \operatorname{PROUD}=0 \text { ! ways } \end{aligned}

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

\begin{aligned} & =4 !+4 !+3 !+3 !+2 !+1+0 ! \\ & =(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)+1+1 \\ & =24+24+6+6+2+1+1 \\ & =64 \end{aligned}

Therefore, the rank of the word PROUD is 64

Thus, the number of words in a dictionary before the word PROUD is 64-1=63

 

Posted by

rishi.raj

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