All the letters of the word G T W E N T Y are written in all possible ways with or without meaning and these…

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

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

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

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

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 $360+60+60+24+24+24+1=553$