The theoretical distributions of a limited amount of energy among small numbers of particles with discrete, evenly-spaced quantum levels are examined systematically. The average populations of energy states reveal the pattern of Pascal's triangle. An exact formula for the probability that a particle will be in any given energy state is derived. This formula, dubbed the Pascal distribution, is compared to the exponential formula for the Boltzmann distribution. Though they differ for systems with small numbers of particles, the Pascal distribution is shown to approach the Boltzmann distribution as the number of particles and total amount of energy increases with a fixed ratio of energy to number of particles. Implications for deriving the Boltzmann distribution from small model systems and for understanding the energy distributions in real nanoscale systems are discussed. (Contains 2 notes, 6 tables and 2 figures.)