Illinois State University Mathematics Department

 MAT 305: Combinatorics Topics for K-8 Teachers Spring 2000 6:00 - 8:50 pm Tuesday STV 332 Dr. Roger Day (day@ilstu.edu)

 Semester Exam possible solutions

Evaluation Criteria

You may earn up to 10 points on each of questions 1 through 10. For each question:

• 6 points count toward a correct solution to the problem. I will evaluate the mathematics you use:
• Is it accurate and appropriate?
• Have you provided adequate justification?
• 4 points count toward how you express your solution. I will evaluate how you communicate your results:
• Is your solution clear and complete?
• Have you expressed logical connections among components of your solution?

1.

 a. Show how to simplify the following expression to generate a positive integer: C(5,2) b. Determine the number of ways to arrange 10 distinct dogs in a straight line. c. There are 8 coffee choices and 12 tea choices on the menu at Farms and Babble Bookstore. These are the only beverage choices. (i) If a customer orders either tea or coffee, how many selections does the customer have to choose from? (ii) If a customer orders one tea choice and one coffee choice, how many choices are possible? Disregard whether tea or coffee is ordered or served first. d. Determine the number of non-negative integer solutions to the equation A + B + C + D + E + F = 40. e. Consider Pascal's Triangle, where 1 is the 0th row, 1 1 is the 1st row, and 1 2 1 is the 2nd row. (i) Show the elements in row 6 of Pascal's Triangle. (ii) State the sum of the elements that appear in row 20 of Pascal's Triangle.

2.

 a. Determine the number of collected terms in the expansion of . b. Determine the value of the coefficient K in the collected term resulting from the expansion of . c. Determine the number of uncollected terms in the expansion of . d. Determine the number of collected terms in the expansion of .

3.

The following conjecture is to be proven true by induction or shown to be false using a counterexample:

 a. State and carry out the first step in the induction process. b. State and carry out the second step in the induction process. c. State but do not carry out the third step in the induction process.

4.

A passenger jet may fly three routes from New York to Chicago and four routes from Chicago to Los Angeles. For a round trip from New York to Los Angeles and back, determine the number of ways a passenger can travel without repeating the same route on any leg of the round trip.

5.

Five unique dice are thrown simultaneously. Determine the portion of all possible throws that results in at least two 5s appearing.

6.

A nurse walks from home at 10th and H to her clinic at 16th and M, always walking to higher numbers or to letters further along in the alphabet. On a certain day, police block off 13th street between K and L streets. What portion of all possible paths from the nurse's home to the clinic contain the blocked-off street?

7.

A company named GAMES has an advertising display with the letters of its name, "GAMES." Colors are used for each letter, but the colors may be repeated. On one particular day, for example, the colors might be red, green, green, blue, red. The company wishes to use a different color scheme for each of the 365 days in the year 2001. Determine the minimum number of colors that are required for this task.

8.

In 7,843 families, all of which have a TV set, a dishwasher, a microwave, and a car, there are six different types of TV sets, five different types of dishwashers, four different types of microwaves, and eight different types of cars. What is the least number of families that have the same type of TV, dishwasher, microwave, and car?

9.

Cannon balls are stacked in a compact equilateral triangular pattern. When there are n layers in the stack, there are n balls per side of the triangle on the lowest layer, n-1 per side on the next layer, and so on, up to 1 ball on the top. Determine a recursion relationship B(n), including any initial conditions, for the total number of balls in a pile with n layers.

10.

Exactly 10 chocolate chips are to be distributed at random into 6 chocolate-chip cookies. What is the probability that some cookie has at least 3 chips in it?

BONUS!

A survey was conducted of 983 families to determine whether they possessed (1) a cell phone, (2) a microwave, (3) a satellite dish, or (4) a CD player. No family was completely without such items, and 481 families had at least two of these items. At least three items were possessed by 345 families and 264 families possessed all 4 items.

 a. Determine the total number of pieces of equipment held by all 983 families. b. Determine the number of families that held the number of pieces of equipment specified below. Assume that none of these families had more than one of any particular item. (i) exactly one piece of equipment (ii) exactly two different pieces of equipment (iii) exactly three different pieces of equipment

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