All the letters of the word GTWENTY are written in all possible...

We have to find the rank of word $GTWENTY$.

Words starting from $E$$= \dfrac{6 !}{2 !} = \dfrac{720}{2} = 360$

Words starting with $GE = \dfrac{5 !}{2} = 60$

Words starting with $GN = \dfrac{5 !}{2} = 60$

Words starting with $GTE = 4 ! = 24$

Words starting with $GTN = 4 ! = 24$

Words starting with $GTT = 4 ! = 24$

Then the next word will be $GTWENTY$ and its rank will be $360 + 60 + 60 + 24 + 24 + 24 + 1 = 553$