Advanced Topics in Algebra
for K-8 Teachers
MAT 305 Spring 1996
Test #3 Possible
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.
You may earn up to 6 points on each of questions 1 through 10. For
--> 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
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
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
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) =
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
7. Consider the expansion of (x + 3y)^5.
(a) Write the collected term that contains the factor
(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
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
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,