We introduce a multi-agent combinatorial contract design problem with type constraints. A principal assigns a task to agents divided into k types. Agents of each type decide whether to exert costly effort. The principal’s reward is a non-negative, monotone, and submodular function of the agents exerting effort across all types. Extending prior work by Dütting et al. (2023) on the single-agent-per-type ( \(k=n\) ) case, we allow multiple agents per type, where n denotes the number of agents. We formulate the contract design as a bi-level optimization problem, which we transform into a single-level subset selection problem using backward induction. We provide a parameterized approximation algorithm using value and demand query oracles, achieving approximation ratios approximately 5 times better than Dütting et al. (2023) under suitable parameter settings.