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 are to work alone on this test. You may not use anyone else'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?
(a) as the product of four consecutive integers;
(b) using factorial notation; and
(c) in terms of C(14,4).
(a) expressed using combination notation; and
(b) expressed as a base-10 number.
NOTE: The set of jokes {A,B,C} is considered one set of jokes, no matter what order Juanita tells the three jokes.
Solution: Think of filling these three
spaces with digits:
As CASE I, suppose the arrangement has no digits greater than 7. We
have 8 digits to choose from for the first position, 7 for the
second, and 6 for the third. We multiply to get 8*7*6 arrangements
with no digits greater than 7.
For CASE II, suppose the arrangement has one digit greater than 7. We have 2 digits to choose from for the digit greater than 7, 8 to choose from for the next digit, and 7 to chose from for the final digit. We multiply to get 2*8*7 arrangements with one digit greater than 7.
The two cases are disjoint, for no arrangement can have no digits greater than 7 and one digit greater than 7 at the same time. Therefore, we add the results of the two cases. There are 8*7*6 + 2*8*7 possible arrangements.
(a) How many collected terms are there?
(b) Write out the first three collected terms, beginning with the
term containing the factor a^5.
(a) How many uncollected terms are there?
(b) How many collected terms are there?
(c) What is the coefficient of the collected term that contains the
factor ef^6g^10h^8