Lambda word problem

Problem You work for a large conglomerate of 10:] associated companies. In order to avoid antitrust issues with the DO]. you need to divide the 100 companies into 4 different groups such that: 1. Companies within the same group cannot do business with each other. 2. Companies in different groups can do business with each other. What organization of the 100 companies into the 4 groups maximimizes the number of business opportunities? a} Start by labeling the groups X, Y, Z. and W. Let :I: = the number of companies assigned to group X and so on for the other groups. Construct an equation in a: for the number of business opportunities For a company in group X i.e.. how many companies can that company do business with? b) Now build an equation in a: for the total number of business opportunities For all companies in group X. c) Do likewise for the remaining groups and construct a function fix, 1;, 2,11!) that gives the total number of business opportunities across all the groups. d} What is the constraint on as, y. z, and w? e} Introduce a Lagrange multiplier )i and determine f: = A9\" where g is the function con- structed from the above constraint. f) Do likewise for f\