Rank of the word DIGITAL?

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$$\begin{gathered} \mathrm{Given} \mathrm{word} \mathrm{Digital} , \\ \mathrm{permutations} \mathrm{with} \mathrm{letter} \mathrm{A} \mathrm{\_} \mathrm{\_} \mathrm{\_} \mathrm{\_} \mathrm{\_} \mathrm{\_} , \mathrm{the} \mathrm{other} 6 \mathrm{letters} \mathrm{can} \mathrm{be} \mathrm{arranged} \mathrm{in} \frac{6!}{2} \mathrm{ways}, \mathrm{since} \mathrm{i} \mathrm{is} \mathrm{repeated}. \\ \Rightarrow \frac{720}{2}=360 \mathrm{ways} \\ \mathrm{permutations} \mathrm{with} \mathrm{letter} \mathrm{Da\_} \mathrm{\_} \mathrm{\_} \mathrm{\_}\hspace{0.17em}\mathrm{\_} , \mathrm{the} \mathrm{other} \mathrm{five} \mathrm{letters} \mathrm{can} \mathrm{be} \mathrm{arranged} \mathrm{in} \frac{5!}{2}=\frac{120}{2}=60 \mathrm{ways} \\ \mathrm{permutations} \mathrm{with} \mathrm{letter} \mathrm{Dg\_} \mathrm{\_} \mathrm{\_} \mathrm{\_} \mathrm{\_} , \mathrm{the} \mathrm{other} \mathrm{five} \mathrm{letters} \mathrm{can} \mathrm{be} \mathrm{arranged} \mathrm{in} \frac{5!}{2}=\frac{120}{2}=60 \mathrm{ways} \\ \mathrm{permutations} \mathrm{with} \mathrm{letter} \mathrm{Dia\_} \mathrm{\_}\hspace{0.17em}\mathrm{\_}\hspace{0.17em}\mathrm{\_}, \mathrm{the} \mathrm{other} 4 \mathrm{letters} \mathrm{can} \mathrm{be} \mathrm{arranged} \mathrm{in} \frac{4!}{2}=\frac{24}{2}=12 \mathrm{ways} \\ \mathrm{permutations} \mathrm{with} \mathrm{letter} \mathrm{Diga\_}\hspace{0.17em}\mathrm{\_}\hspace{0.17em}\mathrm{\_}, \mathrm{the} \mathrm{other} \mathrm{three} \mathrm{letters} \mathrm{can} \mathrm{be} \mathrm{arranged} \mathrm{in} \frac{3!}{2}=\frac{6}{2}=3 \mathrm{ways} \\ \mathrm{permutations} \mathrm{with} \mathrm{letter} \mathrm{Digia\_} \mathrm{\_} , \mathrm{the} \mathrm{other} \mathrm{two} \mathrm{letters} \mathrm{can} \mathrm{be} \mathrm{arranged} \mathrm{in} 2! =2 \mathrm{ways} \\ \mathrm{permutation} \mathrm{with} \mathrm{letter} \mathrm{Digil\_\_} , \mathrm{the} \mathrm{other} \mathrm{two} \mathrm{letters} \mathrm{can} \mathrm{be} \mathrm{arranged} \mathrm{in} 2! =2 \mathrm{ways} \\ \mathrm{the} \mathrm{word} \mathrm{Digital} \mathrm{will} \mathrm{be} \mathrm{the} \mathrm{next} \mathrm{word} . \\ \mathrm{Hence} \mathrm{rank} \mathrm{of} \mathrm{word} \mathrm{digital}=360+60+60+12+3+2+2=500 \\ \mathbf{hence}\mathbf{ }\mathbf{digital}\mathbf{ }\mathbf{will}\mathbf{ }\mathbf{be}\mathbf{ }\mathbf{the}\mathbf{ }\mathbf{501}\mathbf{th}\mathbf{ }\mathbf{word}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{dictionary}\mathbf{ }\mathbf{ } \end{gathered}$$

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