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  • If the letters of the word MASTERFUL are arranged in all possible ways and the words are arranged in a dictionary, then find the rank of the word MASTERFUL.  <div class='qna-opti

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

 

Option: 1

19591


Option: 2

6971


Option: 3

164654


Option: 4

13879


Answers (1)

best_answer

Given that the word is MASTERFUL. 

The lexicographic order of the letters of the given word is A, E, F, L, M, R, S, T, U. The words that begin with A will come first in the lexicographic order.

If the letter A is in the first place of the nine-letter word, then the remaining four letters can be arranged in 8! ways. On proceeding like this, 

8\mathrm{A}(-------)=8 ! \text { ways }

\begin{aligned} & \mathrm{E}(------)=8 ! \text { ways } \\ & \mathrm{F}(-------)=8 ! \text { ways } \\ & L(-------)=8 ! \text { ways } \\ & \operatorname{MAE}(-----)=6 ! \text { ways } \\ & \operatorname{MAF}(-----)=6 ! \text { ways } \\ & \operatorname{MAL}(------)=6 ! \text { ways } \\ & \operatorname{MAR}(-----)=6 ! \text { ways } \end{aligned}

\begin{aligned} & \operatorname{MAR}(-----)=6 ! \text { ways } \\ & \operatorname{MASE}(-----)=5 ! \text { ways } \\ & \operatorname{MASF}(-----)=5 ! \text { ways } \\ & \operatorname{MASL}(-----)=5 ! \text { ways } \\ & \operatorname{MASR}(-----)=5 \text { ! ways } \\ & \operatorname{MASTEF}(---)=3 ! \text { ways } \end{aligned}

MASTEL $(---)=3 !$ \: ways

\operatorname{MASTERFL}(-)=1 ! ways

MASTERFUL $=0$ ! \: ways

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

\begin{aligned} &= 8 !+8 !+8 !+8 !+6 !+6 !+6 !+6 !+5 !+5 !+5 !+5 !+3 !+3 !+1 !+0 ! \\ &=(8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)+(8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)+(8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1) \\ &+(8 \times 7 \times 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)+(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)+1+1 \\ &= 40320+40320+40320+40320+720+720+720+720+120+120+120+120+ \\ & 6+6+1+1 \\ &= 164654 \end{aligned}

Therefore, the rank of the word MASTERFUL is 164654

 

 

Posted by

avinash.dongre

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