Chapter 6: Chapter 6

(Some familiarity with linear programming is assumed for this exercise.) Before the advent of the simplex method for solving linear programming problems, the following method was used: Suppose you have a linear programming problem with three unknowns and 20 constraints. You locate corner points as follows: Selecting three of the constraints, you turn them into equations (by replacing the inequalities with equalities), solve the resulting system of three equations in three unknowns, and then check to see whether the solution is feasible. a. How many systems of three equations in three unknowns will you be required to solve? b. Generalize this to \(n\) constraints.

If you were hard pressed to study for an exam on counting and had only enough time to study one topic, would you choose the formula for the number of permutations or the multiplication principle? Give reasons for your choice.

Explain the meaning of a universal set, and give two different universal sets that could be used in a discussion about sets of positive integers.

The formula for \(C(n, r)\) is written as a ratio of two whole numbers. Can \(C(n, r)\) ever be a fraction and not a whole number? Explain (without actually discussing the formula itself).

Is the set of outcomes when two indistinguishable dice are rolled (Example 1) a Cartesian product of two sets? If so, which two sets; if not, why not?

Which of the following represent permutations? (A) An arrangement of books on a shelf (B) A group of 10 people in a bus (C) A committee of 5 senators chosen from 100 (D) A presidential cabinet of 5 portfolios chosen from 20

Evaluate each number. $$ P(8,3) / 3 ! $$

List the elements in each of the sets. \(B=\\{n \mid n\) is an odd positive integer and \(0 \leq n \leq 8\\}\).

Which of the following represent combinations? (A) A portfolio of five stocks chosen from the S\&P Top Ten (B) A group of 5 tenors for a choir chosen from 12 singers (C) A new company CEO and a new CFO chosen from five (D) The New York Times Top Ten Bestseller list

Rewrite in set notation: She prefers movies that are not violent, are shorter than two hours, and have neither a tragic ending nor an unexpected ending.