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Soft robot serial chain manipulators with the capability for growth, stiffness control, and discrete joints have the potential to approach the dexterity of traditional robot arms, while improving safety, lowering cost, and providing an increased workspace, with potential application in home environments. This paper presents an approach for design optimization of such robots to reach specified targets while minimizing the number of discrete joints and thus construction and actuation costs. We define a maximum number of allowable joints, as well as hardware constraints imposed by the materials and actuation available for soft growing robots, and we formulate and solve an optimization problem to output a planar robot design, i.e., the total number of potential joints and their locations along the robot body, which reaches all the desired targets, avoids known obstacles, and maximizes the workspace. We demonstrate a process to rapidly construct the resulting soft growing robot design. Finally, we use our algorithm to evaluate the ability of this design to reach new targets and demonstrate the algorithm's utility as a design tool to explore robot capabilities given various constraints and objectives.