Multiple sample size approximations and an exact sample size methodology for the audit of nuclear material management are presented. This work is significant because it demonstrates the approximations used in the practice of determining the sample size for an audit of special nuclear material may over-estimate the required sample size causing an unneeded burden and excess resources expended by the auditor or may under estimate the sample size causing incorrect assumptions about the reliability and statistical significance of the audit result. The tables to compute the exact sample sizes for special nuclear material are provided based upon United States Department of Energy (DOE) sampling parameters for physical inventories for a population size up to 15,000 items. This work motivates the exact computation of item sizes by showing exact correspondence to integer number of defects and enforcing the constraint of exactly attaining the minimum detectable defect rate. This is accomplished with a bivariate number of defects mixture distribution representing the population. The number of defects in the bivariate mixture being sequential and bracketing the possibly non-integer defect number obtained by multiplying the minimum detectable defect rate times the population size (expected number of defects). There are allowances described and made for populations which would not have at least one defect based upon the minimum detectable defect rate.