N different numbers arranged at random in a sequence can be considered to be consisting of k groups of numbers alternately in ascending and descending order. Groups of numbers in ascending and descending order are called runs up and down respectively. Kermack and McKendrick1, Levene and Wolfowitz2 and others have dealt with the theory of runs up and down. Recently3, I developed independently a method similar to that of Frechet4 for calculating the factorial moments of a large number of distributions considered in the statistical literature. This method is directly applicable for calculating the factorial and product moments of the distributions arising in the theory of runs up and down. Thus, for example, the second factorial moment for the distribution of the total number of runs of length p, or of p and more, is the sum of the expectations of the different configurations giving two such runs. The expectation for each of the configurations is the product of their probability and the number of configurations that can be had from the N numbers. The probability for any of the configurations can be evaluated step by step by breaking it into two independent sections and expressing it as the sum of two new configurations as illustrated below.