smc

Expand the binomial $((a+b)+c{)}^n$ with the binomial theorem. We have: $$\sum _{k=0}^{10}\left(\genfrac{}{}{0ex}{}{10}{k}\right)(a+b{)}^k{c}^{10-k}$$ So for each iteration of the summation operator, we add k+1 dissimilar terms. Therefore our answer is: $$\sum _{k=0}^{10}k+1=\frac{11(1+11)}{2}=66\to \boxed{66}$$