Hello can you please help me with the first three parts with a step by step guide. Thank you very much! (A,B,and C.) You work for a large conglomerate of 100 associated companies. In order to avoid antitrust issues with the DOJ, 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 x = the number of companies assigned to group X and so on for the other groups. Construct an equation in x for the number of business opportunities for a company in group X — i.e., how many companies can that company do business with? b) Nowbuildanequationinxforthetotalnumberofbusinessopportunitiesforallcompanies in group X. c) Do likewise for the remaining groups and construct a function f (x, y, z, w) that gives the total number of business opportunities across all the groups. d) What is the constraint on x, y, z, and w? e) Introduce a Lagrange multiplier λ and determine fx = λgx, where g is the function con- structed from the above constraint. f) Do likewise for fy, fz, and fw, and combine them with the constraint so that you have 5 equations in 5 unknowns. g) Use substitution to determine a value for λ. h) Substitute the value for λ into the other equations to determine the optimal distribution of companies into the 4 groups.
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
Hello can you please help me with the first three parts with a step by step guide. Thank you very much! (A,B,and C.)
You work for a large conglomerate of 100 associated companies. In order to avoid antitrust issues with the DOJ, 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 x = the number of companies assigned to group X and so on for the other groups. Construct an equation in x for the number of business opportunities for a company in group X — i.e., how many companies can that company do business with?
b) Nowbuildanequationinxforthetotalnumberofbusinessopportunitiesforallcompanies in group X.
c) Do likewise for the remaining groups and construct a function f (x, y, z, w) that gives the total number of business opportunities across all the groups.
d) What is the constraint on x, y, z, and w?
e) Introduce a Lagrange multiplier λ and determine fx = λgx, where g is the function con-
structed from the above constraint.
f) Do likewise for fy, fz, and fw, and combine them with the constraint so that you have 5 equations in 5 unknowns.
g) Use substitution to determine a value for λ.
h) Substitute the value for λ into the other equations to determine the optimal distribution of companies into the 4 groups.
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