Question

Quantos polígonos convexos podem ser formados com os vértices de um octógono

Answer 1

Boa noite!

$\binom{8}{3}+\binom{8}{4}+\binom{8}{5}+\binom{8}{6}+\binom{8}{7}+\binom{8}{8}\\ 2^8-\left(\binom{8}{0}+\binom{8}{1}+\binom{8}{2}\right)\\ 256-\left(\frac{8!}{0!(8-0)!}+\frac{8!}{1!(8-1)!}+\frac{8!}{2!(8-2)!}\right)\\ 256-\left(\frac{8!}{8!}+\frac{8!}{1!7!}+\frac{8!}{2!6!}\right)\\ 256-\left(1+\frac{8\cdot{7!}}{7!}+\frac{8\cdot{7}\cdot{6!}}{1\cdot{2}\cdot{6!}}\right)\\ 256-\left(1+8+\frac{56}{2}\right)\\ 256-\left(1+8+28\right)\\ 256-37=219$

Espero ter ajudado!