When $\left ( 1 - \frac{1}{a} \right ) ^6$ is expanded the sum of the last three coefficients is:
$\textbf{(A)}\ 22\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ -10\qquad\textbf{(E)}\ -11$
This is equivalent to $\frac{(a-1)^6}{a^6}.$
Its expansion has 7 terms, whose coefficients are the same as those of $(a-1)^6$.
By the Binomial Theorem, the sum of the last three coefficients is 
$\binom{6}{2}-\binom{6}{1}+\binom{6}{0}=15-6+1=\boxed{10 \textbf{ (C)}}$.