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

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

 

 

Option: 1

625


Option: 2

1316


Option: 3

4352


Option: 4

15454


Answers (1)

best_answer

Given that the word is TRUSTING.

The lexicographic order of the letters of the given word is G, I, N, R, S, T, T, U. The words that begin with G will come first in the lexicographic order.

If the letter G is in the first place of the eight-letter word, then the remaining four letters can be arranged in 7! ways. On proceeding like this, 

\mathrm{G}(-------)=\frac{7 !}{2 !} \text { ways }

\begin{aligned} & \mathrm{I}(------)=\frac{7 !}{2 !} \text { ways } \\ & \mathrm{N}(-------)=\frac{7 !}{2 !} \text { ways } \\ & \mathrm{R}(-------)=\frac{7 !}{2 !} \text { ways } \\ & \mathrm{S}(-------)=\frac{7 !}{2 !} \text { ways } \\ & \text { TG }(------)=6 ! \text { ways } \\ & \text { TI(------) } 6 ! \text { ways } \\ & \text { TN }(-----)=6 ! \text { ways } \\ & \text { TRG }(-----)=5 ! ways \end{aligned}

\begin{aligned} & \operatorname{TRI}(-----)=5 ! \text { ways } \\ & \text { TRN }(-----)=5 ! \text { ways } \\ & \operatorname{TRS}(-----)=5 ! \text { ways } \\ & \operatorname{TRT}(-----)=5 ! \text { ways } \\ & \operatorname{TRUG}(----)=4 ! \text { ways } \\ & \text { TRUI }(----)=4 ! \text { ways } \\ & \text { TRUN }(----)=4 ! \text { ways } \\ & \operatorname{TRUSG}(---)=3 ! \text { ways } \\ & \text { TRUSI }(---)=3 ! \text { ways } \\ & \text { TRUSN }(---)=3 ! \text { ways } \end{aligned}

TRUSTG $(--)=2 !$ ways

TRUSTIG $(-)=1$ ! ways\\ TRUSTING $=0 !$ ways

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


\begin{aligned} & =\frac{7 !}{2 !}+\frac{7 !}{2 !}+\frac{7 !}{2 !}+\frac{7 !}{2 !}+\frac{7 !}{2 !}+6 !+6 !+6 !+5 !+5 !+5 !+5 !+5 !+4 !+4 !+4 !+3 !+3 !+3 !+2 !+1 !+0 ! \\ & =(7 \times 6 \times 5 \times 4 \times 3)+(7 \times 6 \times 5 \times 4 \times 3)+(7 \times 6 \times 5 \times 4 \times 3)+(7 \times 6 \times 5 \times 4 \times 3)+(7 \times 6 \times 5 \times 4 \times 3) \\ & +(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)+(5 \times 4 \times 3 \times 2 \times 1)+(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)+(3 \times 2 \times 1)+(3 \times 2 \times 1)+(2 \times 1)+1+1 \end{aligned}

\begin{aligned} & =2520+2520+2520+2520+2520+720+720+720+120+120+120+120+120 \\ & +24+24+24+6+6+6+2+1+1 \\ & =15454 \end{aligned}

Therefore, the rank of the word TRUSTING is 15454

Posted by

Irshad Anwar

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