This paper presents an adapted trust-region method for solving computationally expensive black-box optimization problems with mixed binary variables that involve a cyclic symmetry property. Mixed binary problems occur in several practical optimal design problems, e.g., aircraft engine turbines, mooring lines of offshore wind turbines, electric engine stators and rotors. The motivating application for this study is the optimal design of helicopter bladed disk turbo-machines. The necklace concept is introduced to deal with the cyclic symmetry property, and to avoid costly black-box objective function evaluations at equivalent solutions. An adapted distance is proposed for the discrete-space exploration step of the optimization method. A convergence analysis is proposed for the trust-region derivative-free algorithm, DFOb-dH, extended to the mixed-binary case and based on the Hamming distance. The convergence proof is extended to the new algorithm, DFOb-dneck, which is based on the necklace distance. Computational comparison with state-of-the-art black-box optimization methods is performed on a set of analytical problems and on a simplified industrial application.