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  • If the letters of the word QUALITY are arranged in all possible ways and the words are arranged in a dictionary, then find the rank of the word QUALITY.Option: 1

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

Option: 1

2647


Option: 2

6971


Option: 3

3879


Option: 4

23879


Answers (1)

best_answer

Given that the word is QUALITY. 

The lexicographic order of the letters of the given word is A, I, L, Q, T, U, Y. The words that begin with A will come first in the lexicographic order.

If the letter A is in the first place of the seven-letter word, then the remaining four letters can be arranged in 6! ways. On proceeding like this

$$ A(------)=6 \text { ! } $$ ways

1(-----)=6 \text { ! } $$ ways

\begin{aligned} & \mathrm{L}(----)=6 ! \text { ways } \\\ & \text { QA }(----)=5 ! \text { ways } \\ & \text { QI }(----)=5 ! \text { ways } \\ & \text { QL }(-----)=5 ! \text { ways } \\ & \text { QT }(-----)=5 \text { ! ways } \\ & \text { QUAI }(---)=3 ! \text { ways } \\ & \text { QUALITY }=0 \text { ! ways } \end{aligned}

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

\begin{aligned} = & 6 !+6 !+6 !+5 !+5 !+5 !+5 !+3 !+0 ! \\ = & (6 \times 5 \times 4 \times 3 \times 2 \times 1)+(6 \times 5 \times 4 \times 3 \times 2 \times 1)+(6 \times 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) \\ & +(3 \times 2 \times 1)+1 \\ = & 720+720+720+120+120+120+120+6+1 \\ = & 2647 \end{aligned}

the rank of the word QUALITY is 2647

 

Posted by

Sanket Gandhi

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