We propose a method to estimate population size based on capture-recapture designs of K samples. The observed data is formulated as a biased sample of n iid K-dimensional vectors of binary indicators from a conditional distribution given the vector is not 0, where the k-th component indicates that subject being caught by the k-th sample. The target quantity is the prob. that the vector is not 0. We cover models assuming a single general constraint on the K-dimensional distribution so that the target quantity is identified and the statistical model is unrestricted. We present worked out solutions for common constraints (K-way additive interaction=0 and conditional independence). We show the dramatic impact of the choice of constraint on the estimand value, so it’s crucial for the constraint to hold by design. For the K-way multiplicative interaction=0 constraint, MLE suffers from the curse of dimensionality. We propose a targeted MLE that combines machine learning to smooth across the 2^K cells while targeting the fit towards the target parameter of interest. For each problem, we provide simulations wrt assumption violations, inference with CI and experimental designs with software.