If the letters of the word ^ ' MOTHER ^ ' be permuted and all the...

$'{MOTHER}^{'}$

Rearrange in alphabetic order $EHMORT$

Number of words start with $E = 5 !$

Number of words start with $H = 5 !$

Number of words start with $ME = 4 !$

Number of words start with $MH = 4 !$

Number of words start with $MOE = 3 !$

Number of words start with $MOH = 3 !$

Number of words start with $MOR = 3 !$

Number of words start with $MOTE = 2 !$

Number of word $MOTHER = 1$

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