In a Maryland county, 180 baseball fans have been surveyed about the baseball teams they watch on TV. 85 fans watch the Washington Nationals, 127 fans watch the Baltimore Orioles and 162 watch the Washington Nationals or the Baltimore Orioles or both.
(a) How many of the fans watch both the Washington Nationals and the Baltimore Orioles? Show work.
(b) How many of the fans watch the Baltimore Orioles and not the Washington Nationals? Show work.
(c) Complete the following Venn diagram, filling in the number of fans belonging in each of the four regions. Circle W = {fans who watch Washington Nationals} and Circle B = {fans who watch Baltimore Orioles}. (no explanation required)
2. (25 points)
A company makes two raisin-nut mixtures. Each box of mixture A contains 4 ounces of peanuts and 3 ounce of raisins, and sells for $4. Each box of mixture B contains 2 ounces of peanuts and 6 ounces of raisins, and sells for $6. The company has available 2,000 ounces of peanuts and 3,300 ounces of raisins, and will try to sell the amount of each mixture that maximizes income. If x is the number of boxes of mixture A and y is the number of boxes of mixture B, list the inequalities that must be satisfied and state the objective function. Obtain x and y that maximizes the objective function.
(a) Tabulate the given data as follows:
Per Box A Per Box B Available amount (oz) (total)
Peanuts (oz)
Raisins (oz)
Income
(b) State an expression for the total income J from selling x boxes of mixture A and y boxes of mixture B.
(c) Using the data in (a), state two inequalities that x and y must satisfy because of the company’s total availability limits of peanuts and raisins. Show work.
(d) State two inequalities that x and y must satisfy because they cannot be negative.
(e) State the linear programming problem which corresponds to the situation described. Be sure to indicate whether you have a maximization problem or a minimization problem, and state the objective function and all the inequalities. (This part is mostly a summary of the previous parts)
(f) Solve the linear programming problem. You will need to find the feasible region and determine the corner points. You do not have to submit your graph, and you do not have to show algebraic work in finding the corner points, but you must list the corner points of the feasible region and the corresponding values of the objective function. Show work in evaluating J in the following table:
Corner Point (x, y) Value of Objective Function J
(g) Write your conclusion with regard to the word problem. State how many boxes of mixture A and how many boxes of mixture B should be sold, in order to earn the highest income possible. State the value of that maximum income.
3. (10 points)
Using the digits 0, 1, 2, 3, 4, 5, 6, 7 and 8, all possible 3 digit numbers are formed, excluding 0 as the first digit and assuming the digits may be repeated in each number.
(a) How many of these numbers are odd? Show work.
(b) How many of these numbers are even? Show work.
4. (5 points)
Use the given information to complete the following table.
n(U) = 60 , n(A) = 23, n(B) = 20, n(A B) = 6. (No work/explanation required)
A A Totals
B
B
Totals
5. (10 points)
An advisory board of 5 students is to be chosen from a group of 13 students.
(a) In how many ways can the advisory board be chosen? Show work/explanation.
(b) Now suppose that the group of students consists of 7 seniors, 4 juniors, and 2 sophomores. In how many ways can the advisory board be chosen if it must consist of 2 seniors, 2 juniors, and 1 sophomore? Show some work/explanation.
6. (25 points)
An urn contains 25 balls of which 15 are red 10 are white. A sample of 5 balls is to be selected.
(a) How many different samples are possible?
(b) How many samples have all red balls?
(c) How many samples contain 3 red balls and 2 white balls?
(d) How many samples contain at least 4 red balls?
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