All the arrangements,with or without meaning, | vedclass

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Question

(B) The letters of the word $FARMER$ are $A, E, F, M, R, R$. Total letters = $6$. The number of arrangements where two $R$s are together is calculated

Vedclass · Accepted Answer

$77$

Vedclass · Answer

(B) The letters of the word $FARMER$ are $A, E, F, M, R, R$. Total letters = $6$. The number of arrangements where two $R$s are together is calculated by treating $RR$ as one unit. Total arrangements of $FARMER$ is $\frac{6!}{2!} = 360$. Arrangements with $RR$ together is $5! = 120$. So,total valid arrangements = $360 - 120 = 240$.To find the rank of $FARMER$ in dictionary order excluding words with $RR$ together:$1$. Words starting with $A$: $\frac{5!}{2!} - 4! = 60 - 24 = 36$.$2$. Words starting with $E$: $\frac{5!}{2!} - 4! = 60 - 24 = 36$.$3$. Words starting with $FA...$:- $FAE...$: $3! = 6$.- $FAM...$: $3! = 6$.- $FAR...$: We need to arrange $E, M, R$. Total $3! = 6$. Words with $RR$ together is not possible here as only one $R$ is left. So $6$ words.- $FARE...$: $2! = 2$.- $FARM...$: $2! = 2$.- $FARMER$: $1$.Summing up: $36 + 36 + 1 + 1 + 1 + 1 + 1 = 77$.

All the arrangements,with or without meaning,of the word $FARMER$ are written excluding any word that has two $R$ appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word $FARMER$ in this list is .... .

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