Advanced Topics in Algebra
for K-8 Teachers
MAT 305 Spring 1996
Test #3
Test #3 Possible Solutions

Please write your solutions on one side only of each piece of paper you use. You may use factorial notation as well as combination and permutation notation unless instructed otherwise.

You may work in groups of no more than four on this test. You may not use any other group's work nor may you refer to any materials as you complete the test, other than those provided with the test. You may ask me questions.

Evaluation Criteria

You may earn up to 6 points on each of questions 1 through 10. For each question:

--> 4 or 5 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?

--> 1 or 2 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. Consider the expansion of (a + b)^10.

(a) Determine the coefficient K of the collected term Ka^7b^3.

(b) After collecting all terms, what is the sum of the coefficients?

2. Write a recursive representation for the relationship described by the sequence of values 1, 2, 4, 8, 16, . . . .

3. Consider the word PARAMATTA.

(a) Assuming identical letters are indistinguishable, how many distinct rearrangements exist for the letters in this word?

(b) How many unique ways can the letters be rearranged so that no two consonants are adjacent, assuming again that identical letters are indistinguishable?

4. List every unique term generated by the recursion relation t(n)=[5t(n-1)]/[t(n-2)] , where t(0) = 5 and t(1) = 10.

5. Jannier Plinckton arranges the rotary mower display at Biltmore's Toro on main street. Jannier claims that he has enough mowers for display so that he can create a unique line-up of rotary mowers for each day of the entire Spring Sale Season, when Biltmore's is open 6 days a week March through May. What is the fewest number of mowers Jannier must have for display?

6. Determine the number of nonnegative integer solutions to the equation Sigma(k,1,24;x(k))=8.

7. Consider the expansion of (x + 3y)^5.

(a) Write the collected term that contains the factor x^2y^3

(b) Determine an ordered pair (x,y) such that (x + 3y)^5 = 0.

8. Nat Nightwatcher works the evening shift for the Illinois Department of Transportation. Her assignment last month was to record the type of vehicle using Veteran's Parkway from midnight to 6 am each day. She recorded whether or not each of five types of motorized vehicles were present during that time period. The vehicle types she watched for included car, bus, emergency vehicle, motorcycle, and truck. Each morning at 8 am she called in a report to Springfield indicating which of the five vehicle types had been spotted.

Write a brief argument stipulating whether or not Nat could have submitted a different report each morning during last month's assignment.

9. Norton Morton has played a musical instrument in the Melton Memorial Marching Musicians for 25 straight summers. During those summers, he claims to have played the trumpet, the trombone, and the tuba. In fact, he remembers he has played each of those instruments in no less than 6 years of service to the band.

If Norton never played more than one instrument in any one summer and he only played the instruments indicated above, how many different possibilities exist for the number of summers he's played each instrument?

10. The numbers 1, 5, 12, 22, 35, 51, 70, 92, . . . , are called "pentagonal numbers" because these are the numbers of dots that can be arranged in regular pentagons, as shown here.

Looking at the sketch, notice that the fourth pentagonal number is formed from the third one by adding three rows of dots, one containing four dots and the other two containing three dots each. In general, if P(n) is the nth pentagonal number, then P(n+1) = P(n) + (n + 1) + 2n = P(n) + (3n + 1).

Use an induction proof to show that for all positive integers n, P(n)=(n/2)(3n-1).

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