11 + a^ ( n - m ) + 11 + a^ ( m - n ) = ?

Answer: Option C

$$\begin{array}{rl}& \frac{1}{1+a^{\left(n-m\right)}}+\frac{1}{1+a^{\left(m-n\right)}}\\ & =\frac{1}{\left(1+\frac{a^n}{a^m}\right)}+\frac{1}{\left(1+\frac{a^m}{a^n}\right)}\\ & =\frac{a^m}{\left(a^m+a^n\right)}+\frac{a^n}{\left(a^m+a^n\right)}\\ & =\frac{\left(a^m+a^n\right)}{\left(a^m+a^n\right)}\\ & =1\end{array}$$