[tex]{A ^ {3} _{9}} + \frac{{A ^ {4} _{8}}}{{A ^ {3} _{7}}} = \frac{9!}{(9 - 3)!} + \frac{ \frac{8!}{(8 - 4)!}}{ \frac{7!}{(7 - 3)!}}[/tex]
[tex] = \frac{1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9}{6!} + \frac{ \frac{1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8}{4!}}{ \frac{1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7}{4!}}[/tex]
[tex] = \frac{1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9}{1 \times 2 \times 3 \times 4 \times 5 \times 6} + \frac{1 \times 2 \times 3 \times 4 \times 5 \times 6 \times7 \times 8 }{1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7} [/tex]
[tex] = 7 \times 8 \times 9 + 8[/tex]
[tex] = 504 + 8[/tex]
[tex] = 512[/tex]
