In this article, we introduce a novel univariate discrete distribution called the r-hypergeometric model. This distribution has the sampling characteristics with replacement of the binomial distribution and no-order sampling of the hypergeometric distribution. Mathematical expressions are obtained for computing probabilities, the mode, and moments of the new distribution. Two simulation algorithms are proposed using the acceptance-rejection and inverse-transform methods and computational features to generate values of an r-hypergeometric distributed random variable. Python and R codes are implemented to perform computational experiments. Applications of mathematical methods based on the new distribution in sciences and engineering employing simulated and real-world data sets are provided. A comparison with existing distributions is also included. In addition to the mathematical results, some findings obtained from our study are related to a better computational performance of the inverse-transform method. Also, we identify applications of our model that are not covered by the traditional count distributions. In addition, we establish distinct probabilities for the same event under the binomial, hypergeometric, and r-hypergeometric distributions, with their means also being distinct, but their variances, skewness, and kurtosis converge to the same value.