All the letters of the word $$GTWENTY$$ are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word $$GTWENTY$$ is _______

Alphabetical order of letters: E, G, N, T, T, W, Y. (Note: T is repeated twice).

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

Words starting with G: We now look for the word GTWENTY.

o Next letter is E: $$G E \dots$$ remaining (N, T, T, W, Y) $$\rightarrow \frac{5!}{2!} = 60$$

o Next letter is N: $$G N \dots$$ remaining (E, T, T, W, Y) $$\rightarrow \frac{5!}{2!} = 60$$

o Next letter is T: $$G T \dots$$

$$G T E \dots$$ remaining (N, T, W, Y) $$\rightarrow 4! = 24$$

$$G T N \dots$$ remaining (E, T, W, Y) $$\rightarrow 4! = 24$$

$$G T T \dots$$ remaining (E, N, W, Y) $$\rightarrow 4! = 24$$

$$G T W \dots$$ (This matches our word)

$$G T W E \dots$$

$$G T W E N \dots$$

$$G T W E N T Y$$ (This is the first word in this sequence) $$\rightarrow 1$$

Total Rank: $$360 + 60 + 60 + 24 + 24 + 24 + 1 = \mathbf{553}$$