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

 

Option: 1

5783


Option: 2

4597


Option: 3

5643


Option: 4

3690


Answers (1)

best_answer

Given that the word is VIBRANT. 

The lexicographic order of the letters of the given word is A, B, I, N, R, T, V. The words that begin with A will come first in the lexicographic order.

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

\begin{aligned} & \mathrm{A}(-----)=6 ! \text { ways } \\ & \mathrm{B}(-----)=6 ! \text { ways } \\ & \mathrm{I}(-----)=6 ! \text { ways } \\ & \mathrm{N}(------)=6 ! \text { ways } \\ & \mathrm{R}(------)=6 ! \text { ways } \\ & \mathrm{T}(------)=6 ! \text { ways } \\ & \mathrm{VA}(-----)=5 ! \text { ways } \\ & \text { VB }(----)=5 \text { ! ways } \\ & \mathrm{VIA}(----)=4 ! \text { ways } \\ & \mathrm{VIBA}(---)=3 ! \text { ways } \\ \end{aligned}

\operatorname{VIBN}(---)=3 ! ways\\

\mathrm{VIBRANT}=0 ! ways

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

\begin{aligned} & =6 !+6 !+6 !+6 !+6 !+6 !+5 !+5 !+4 !+3 !+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)+(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) \\ & +(4 \times 3 \times 2 \times 1)+(3 \times 2 \times 1)+(3 \times 2 \times 1)+1 \\ & =720+720+720+720+720+720+120+120+24+6+6+1 \\ & =4597 \end{aligned}

Therefore, the rank of the word VIBRANT is  4597

 

Posted by

Divya Prakash Singh

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