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

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

 

Option: 1

625


Option: 2

309


Option: 3

380


Option: 4

964


Answers (1)

best_answer

Given that the word is AMAZING. 

The lexicographic order of the letters of the given word is A, A, I, G, M, N, Z. 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 in5! ways. On proceeding like this,

\begin{aligned} & \mathrm{AA}(-----)=5 ! \text { ways } \\ & \mathrm{Al}(-----)=5 ! \text { ways } \\ & \mathrm{AG}(-----)=5 ! \text { ways } \\ & \mathrm{AMAI}(---)=3 ! \text { ways } \\ & \mathrm{AMAG}(---)=3 ! \text { ways } \\ & \mathrm{AMAN}(---)=3 ! \text { ways } \\ & \mathrm{AMAZIG}(-)=1 ! \text { ways } \\ & \mathrm{AMAZING}=0 ! \text { ways } \end{aligned}

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

\begin{aligned} = & 5 !+5 !+5 !+3 !+3 !+3 !+1 !+0 ! \\ & =(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) \\ & +(3 \times 2 \times 1)+(3 \times 2 \times 1)+1+1 \\ = & 120+120+120+6+6+6+1+1 \\ = & 380 \end{aligned}

Therefore, the rank of the word AMAZING is  380

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Pankaj

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