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If the letters of the word IDEAL 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 IDEAL.

Option: 1

63


Option: 2

89


Option: 3

84


Option: 4

80


Answers (1)

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Given that the word is IDEAL. 

The lexicographic order of the letters of the given word is A, D, E, I, L. 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{D}(----)=4 ! \text { ways } \\ & \mathrm{E}(----)=4 ! \text { ways } \\ & \mathrm{IA}(---)=3 ! \text { ways } \\ & \text { IDA }(--)=2 ! \text { ways } \\ & \text { IDEAL }=0 ! \text { ways } \end{aligned}

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

\begin{aligned} & =4 !+4 !+4 !+3 !+2 !+0 ! \\ & =(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)+(2 \times 1)+1 \\ & =24+24+24+6+2+1 \\ & =81 \end{aligned}

Therefore, the rank of the word IDEAL is 81

Thus, the number of words in a dictionary before the word IDEAL is 81-1=80

Posted by

sudhir.kumar

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