Suppose that there are n jobs and n machines and it costs c_ij to execute job i on machine j. The assignment problem concerns the determination of a one-to-one assignment of jobs onto machines so as to minimize the cost of executing all the jobs. The average case analysis of the classical random assignment problem has received a lot of interest in the recent literature, mainly due to the following pleasing conjecture of Parisi: The average value of the minimum-cost permutation in an n × n matrix with i.i.d. exp(1) entries equals \sum\nolimits_{i = 1}^n {\frac{1}{{i^2 }}}. Coppersmith and Sorkin (1999) have generalized Parisi?s conjecture to the average value of the smallest k-assignment when there are n jobs and m machines. We prove both conjectures based on a common set of combinatorial and probabilistic arguments.