$\left({ }^8{\mathrm{C}}_1{-}^8{\mathrm{C}}_2{+}^8{\mathrm{C}}_3{-}^8{\mathrm{C}}_4{+}^8{\mathrm{C}}_5{-}^8{\mathrm{C}}_6{+}^8{\mathrm{C}}_7{-}^8{\mathrm{C}}_8\right)$ equals:

Let $\mathrm{A}{=}^8{\mathrm{C}}_1{-}^8{\mathrm{C}}_2{+}^8{\mathrm{C}}_3{-}^8{\mathrm{C}}_4{+}^8{\mathrm{C}}_5{-}^8{{{\mathrm{C}}_6}^{+}}^8{\mathrm{C}}_7{-}^8{\mathrm{C}}_8$

$=\frac{8!}{1!7!}-\frac{8!}{2!6!}+\frac{8!}{3!5!}-\frac{8!}{4!4!}+\frac{8!}{5!3!}-\frac{8!}{6!2!}+\frac{8!}{7!1!}-\frac{8!}{0!8!}$

Note: ${ }^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}}=\frac{\mathrm{n}!}{\mathrm{r}!(\mathrm{n}-\mathrm{r})!}$

Thus, $A=8-\frac{8\times 7}{2}+\frac{8\times 7\times 6}{3\times 2}-\frac{8\times 7\times 6\times 5}{4\times 3\times 2\times 1}+\frac{8\times 7\times 6}{3\times 2}-\frac{8\times 7}{2}+8-1$

and $A=8-28+56-70+56-28+8-1=1$