3.1-3.3, 4.1, 6.1-6.4
Circle the correct choice.
U = {A, B, c,D,e, 0, 1, 2, 3, 4}, F = {A, B, c, D}, G = {c, e, 0, 1, 2}, and H = {0, 1, 2, 3, 4}
True False 1. bF.
True False 2. G has 5 subsets.
True False 3. FH = { }.
True False 4. {xU | x is a letter in the word BAD} HC.
True False 5. (FH)C = U
6. You are given an exam of 5 true/false questions and 8 multiple choice questions, each with 5 possible answers. How many ways can the exam be answered, if no question is left unanswered?
A. 560
B. 3,840
C. 134,400
D. 12,500,000
E. none of these
7. Stephanie’s piggy bank has 8 pennies, 9 nickels, 6 dimes, and 5 quarters. In how many ways can she choose 4 coins from her piggy bank and end up with at most 1 quarter?
A. 17,710
B. 8,855
C. 2,160
D. 432
E. none of these
8. There are five freshmen, four sophomores, six juniors, and two seniors. How many ways could they all sit in a row if each grade has to sit together as a group?
A. 49,766,400
B. 4,147,200
C. 99,532,800
D. 240
E. none of these
9. Stephanie’s piggy bank has 8 pennies, 9 nickels, 6 dimes, and 5 quarters. In how many ways can she choose 4 coins from her piggy bank and end up with exactly 3 dimes?
A. 3,000
B. 7,200
C. 440
D. 500
E. none of these
10. An eight card hand is dealt from a standard deck of 52 cards. In how many ways can this be done if there are to be exactly 3 diamonds, 2 clubs, and 2 hearts?
A. 62,748,517
B. 41,760,576
C. 22,620,312
D. 1,740,024
E. none of these
11. How many 7 letter code words can be created from 2 letters and then 5 digits if no letter or digit can be repeated?
A. 67,600,000
B. 19,656,000
C. 32,500
D. 81,900
E. none of these
12. SET UP (but DO NOT SOLVE) the following linear programming problem (8 pts):
The owner of a health store wishes to prepare a low-calorie fruit drink with a high vitamin A and C content by blending cranberry juice and orange juice. Each glass of the blend is to contain at least 1200 units of vitamin A and 200 units of vitamin C. One ounce of cranberry juice contains 60 units of vitamin A, 16 units of vitamin C, and 14 calories. One ounce of orange juice contains 120 units of vitamin A, 12 units of vitamin C, and 11 calories. The drink must contain at least twice as much cranberry juice as orange juice. How many ounces of each juice should the blend contain to meet the vitamin requirements if the number of calories is to be maximized?
Solution: Maximize P=14x + 11y Subject to: 60x + 120y 1200
16x + 12y 200
x 2y
x, y 0
13. For the given linear programming problem, set up the initial simplex tableau (8 pts).
Maximize P = 3x + 2y–zSolution:
Subject to: 3y + 2x – 3z 16
2x – 3y 14
x 0, y 0, z 0
14. For the given simplex tableau, circle the first pivot element, showing work supporting your decision (4 pts).
Solution: -4 is most negative; 3 is least nonnegative
15. For the given simplex tableau, list the basic and nonbasic variables and give their values (8 pts).
16. Is the solution in #15 the optimal solution (2 pts)? YES or NO (circle one)
17. A survey of 150 sports fans was conducted.
F is the set of people surveyed who like football.
S is the set of people surveyed who like soccer.
B is the set of people surveyed who like basketball.
Use the information given to fill in the provided Venn diagram (8 pts).
- 72 like football
- 61 like soccer
- 10 like only basketball
- 18 like football and basketball
- 5 like soccer and basketball, but not football
- 12 like soccer and basketball
- 28 like football and soccer
18. Using set notation (F, S, B, , , C), describe the set of people surveyed (see #17) who like only soccer (2 pts).
Solution: S FC BC
19. Shade the portion of the Venn diagram that represents the given set (4 pts).
(AB) CC
20. Shade the portion of the Venn diagram that represents the given set (4 pts).
B(ACC)
- Graph the problem and use the Method of Corners to solve the problem (8 pts).
Maximize: P = 3x – 2y
Subject to: 3x – y -2
x + y 6
x 0, y 0
Solution: