If the letters of the word ' MOTHER ' be permuted and all the words so formed (with or without…

${}^{\text{'}}{\mathrm{MOTHER}}^{\text{'}}$

Rearrange in alphabetic order $\mathrm{EHMORT}$

Number of words start with $\mathrm{E}=5!$

Number of words start with $\mathrm{H}=5!$

Number of words start with $\mathrm{ME}=4!$

Number of words start with $\mathrm{MH}=4!$

Number of words start with $\mathrm{MOE}=3!$

Number of words start with $\mathrm{MOH}=3!$

Number of words start with $\mathrm{MOR}=3!$

Number of words start with $\mathrm{MOTE}=2!$

Number of word $\mathrm{MOTHER}=1$

$2\times 5!+2\times 4!+3\times 3!+2!+1=240+48+18+2+1=309$