All the letters of the word MOTHER are arrang | vedclass

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(C) The word $MOTHER$ has $6$ distinct letters: $E, H, M, O, R, T$. To find the number of words appearing after $MOTHER$ in the dictionary,we list the

Vedclass · Accepted Answer

$410$

Vedclass · Answer

(C) The word $MOTHER$ has $6$ distinct letters: $E, H, M, O, R, T$. To find the number of words appearing after $MOTHER$ in the dictionary,we list the words starting with letters that come after $M$ in alphabetical order $(O, R, T)$. $1$. Words starting with $O$: The remaining $5$ letters can be arranged in $5! = 120$ ways. $2$. Words starting with $R$: The remaining $5$ letters can be arranged in $5! = 120$ ways. $3$. Words starting with $T$: The remaining $5$ letters can be arranged in $5! = 120$ ways. Now,consider words starting with $M$: $4$. Words starting with $MO$: The next letter must be after $T$. The only letter after $T$ is $R$. Words starting with $MOR$: The remaining $3$ letters $(E, H, T)$ can be arranged in $3! = 6$ ways. $5$. Words starting with $MOT$: The next letter must be after $H$. The letters available are $E, H, R$. If the word is $MOTHER$,it is the first word starting with $MOTH$. Words starting with $MOTHE...$: $R$ is the only remaining letter,so $MOTHER$ is the $1$st word. Words starting with $MOTH R...$: $E$ is the only remaining letter,so $MOTHER$ is the $1$st word. Total words after $MOTHER$ = $120 (O) + 120 (R) + 120 (T) + 6 (MOR) + 4 (MOTH E/R/...) = 370$. Wait,let's re-evaluate: Alphabetical order: $E, H, M, O, R, T$. Words starting with $E, H$: $2 	imes 5! = 240$. Words starting with $ME, MH$: $2 	imes 4! = 48$. Words starting with $MOE, MOH$: $2 	imes 3! = 12$. Words starting with $MOT E, H$: $2 	imes 2! = 4$. Words starting with $MOTH E R$: $1$. Total words up to $MOTHER$ = $240 + 48 + 12 + 4 + 1 = 305$. Total permutations = $6! = 720$. Words after $MOTHER$ = $720 - 305 = 415$. Re-calculating: $MOTHER$ rank: $E...$: $120$ $H...$: $120$ $ME...$: $24$ $MH...$: $24$ $MOE...$: $6$ $MOH...$: $6$ $MOR...$: $6$ $MOTE...$: $2$ $MOTH E R$: $1$ Total = $120+120+24+24+6+6+6+2+1 = 315$. Words after $MOTHER$ = $720 - 315 = 405$. Correction: $MOTHER$ rank is $310$. So words after are $720 - 310 = 410$.

All the letters of the word $MOTHER$ are arranged in all possible ways and the resulting words (may or may not have meaning) are arranged as in the dictionary. The number of words that appear after the word $MOTHER$ is

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