Question

If the letters of the word 'MOTHER' are written in all possible orders and these words are written out as in a dictionary, find the rank of the word 'MOTHER'.

Answer 1

MOTHER

Letters in alphabetical $\to E,H,M,O,R,T$

$\therefore$ no of 6 letters woed starting with $E\Rightarrow E............\Rightarrow 5!$ ways

no of 6 letter words starting with $H\Rightarrow H.........\Rightarrow 5!$ ways

no of 6 letter words starting with $ME\Rightarrow ME.......4!$ ways

no of 6 letter words starting with $MH\Rightarrow MH.......4!$ ways

no.of 6 letters words starting with $MOE\Rightarrow MOE.....3!$ ways

no.of 6 letters words starting with $MOH\Rightarrow MOH.....3!$ ways

no.of 6 letters words starting with $MOR\Rightarrow MOR.....3!$ ways

no.of 6 letters words starting with $MOTE\Rightarrow MOTE.....2!$ ways

no.of 6 letters words starting with $MOTHE\Rightarrow MOTHER.....1$

$\therefore$ rank of mother $\Rightarrow 5!+5!+4!+4!+3!+3!+3!+2!+1$

$\Rightarrow 2(5!)+2(4!)+3(3!)+2+1$

$\Rightarrow 240+48+18+3$

$\Rightarrow 309$

$\therefore$ Rank of MOTHER is 309