In this chapter, we offer a simple yet powerful approach in which the independent variables 1,…, n in multiple symmetric functions and Vieta’s formulae are replaced by the indicator functions of the events Ai, I = 1,…, n, I = 1(Ai), I = 1,…, n. Both the random variable K, which counts the number of real occurrences, and the suggested identity ni =1(z 1(Ai)) (z 1)KznK, which is completely dependent on K, are important in this chapter. We may offer further easy proofs of established discoveries as well as get new conclusions by picking alternative values for z (real, complex, and random) and obtaining expectancies of the various functions. The expected elementary symmetric functions are computed using the estimated procedures. In the complex domain (z C), least squares based on IFFT and least squares or linear programming in the real domain (z R) are notable. To generate novel outcomes and inequalities, we employ Newton’s identities and a few well-known inequalities. For finite sample spaces, we describe a method that computes the distribution of K (i.e., qk:= P(K = k), k = 0, 1,…, n). Finally, we provide some recommendations for further study.
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