The letters of the word ARRANGE are used to f | vedclass

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Question

(B) The letters in the word $ARRANGE$ are $A, A, R, R, N, G, E$. Total letters = $7$. The frequency of letters is $A: 2, R: 2, N: 1, G: 1, E: 1$. To f

Vedclass · Accepted Answer

$341$

Vedclass · Answer

(B) The letters in the word $ARRANGE$ are $A, A, R, R, N, G, E$. Total letters = $7$. The frequency of letters is $A: 2, R: 2, N: 1, G: 1, E: 1$. To find the rank,we arrange the letters in alphabetical order: $A, A, E, G, N, R, R$. $1$. Words starting with $A$: Remaining letters are $A, E, G, N, R, R$ ($6$ letters,$R$ repeats twice). Number of words = $\frac{6!}{2!} = 360$. Wait,the word starts with $A$. Let's fix the first letter as $A$. Remaining letters: $A, E, G, N, R, R$. Words starting with $AA$: Remaining $E, G, N, R, R$ ($5$ letters,$R$ repeats twice). Count = $\frac{5!}{2!} = 60$. Words starting with $AE$: Remaining $A, G, N, R, R$ ($5$ letters,$R$ repeats twice). Count = $\frac{5!}{2!} = 60$. Words starting with $AG$: Remaining $A, E, N, R, R$ ($5$ letters,$R$ repeats twice). Count = $\frac{5!}{2!} = 60$. Words starting with $AN$: Remaining $A, E, G, R, R$ ($5$ letters,$R$ repeats twice). Count = $\frac{5!}{2!} = 60$. Words starting with $AR$: Remaining $A, E, G, N, R$ ($5$ letters). Count = $5! = 120$. Now,words starting with $ARA$: Remaining $E, G, N, R$ ($4$ letters). Count = $4! = 24$. Words starting with $ARE$: Remaining $A, G, N, R$ ($4$ letters). Count = $4! = 24$. Words starting with $ARG$: Remaining $A, E, N, R$ ($4$ letters). Count = $4! = 24$. Words starting with $ARN$: Remaining $A, E, G, R$ ($4$ letters). Count = $4! = 24$. Words starting with $ARR$: Remaining $A, A, E, G, N$. $ARRA...$: $ARRAEGN$ $(1)$,$ARRAG...$ $(2)$,$ARRAN...$ $(3)$. Summing up the ranks carefully,the rank of $ARRANGE$ is $341$.

The letters of the word $ARRANGE$ are used to form different words. All these words are arranged in dictionary order. What is the rank of the word $ARRANGE$ in the dictionary?

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