At least 1 solution

Lets pick numbers 1 and 0

A. LHS -(2)^0 = -1 RHS (-2)^-n = 1/1 = 1 (No SOLUTION)
B. LHS 2^(-n) = 2^0 = 1 RHS (-2)^n = (-2)^0 = 1 VALID SOLUTION
C. LHS 2^n = 2^0 = 1 RHS (-2)^-n = (-2)^0 = 1 VALID SOLUTION
D. LHS (-2)^n = (-2)^1 = -2 RHS -2^n = -2^1 = -2 VALID SOLUTION
E. LHS (-2)^-n = (-2)^-1 = -(1/2) RHS -2^-n = -(2^-1) = -(1/2) VALID SOLUTION

Hence Only A does not have any valid solution.
Ans A