MATH 387
Final Exam Solutions
1. (22 points) Below, a number is any string of digits that does not begin with a zero.
(a) (2 points) How many 6-digit numbers are there?
We may select the rst digit in any of 9 ways (any digit from 19), and the remaining
ve
MATH 387-01
Practice Exam #1 Solutions
1. You nd that you need to buy 22 hats. The hat shop has as many hats as you might desire in
four dierent varieties: stetsons, berets, stovepipes, and pillboxes. Hats within a single variety
are identical.
(a) How ma
MATH 387-01
Exam #1 Solutions
1. (25 points) You are asked to assign your six subordinates (Alice, Bob, Carla, Dave, Ed, and
Fiona) to 3 specic projects (codenamed Runcible, Screaming Fist, and Valis).
(a) (5 points) How many ways are there to do this if
MATH 387-01
Problem Set #1
Learning to Count
In this section you should use a methodical complete list of examples to determine the answer
the question; even if you know other, non-listing techniques to answer the question, please
explicitly list the obje
MATH 387-01
Problem Set #2
Finding and using appropriate statistics
You can use either a standard combinatorial statistic (exponents, binomials, multinomials,
etc.), or a combination of multiple combinatorial statistics to solve these problems; explain
wh
MATH 387-01
Practice Exam #2 Solutions
Name:
1. (15 points) Computationally, a vector is simply a list of numbers. We may represent an
n-dimensional vector a as a list of n coordinates (a1 , a2 , a3 , . . . , an ).
(a) (10 points) Write an algorithm, usin
MATH 387-01
Problem Set #4
1. (6 points) Find (but do not solve) a homogeneous linear recurrence relation, including
initial conditions, for the number an of ways to tile a 2 n rectangle with any number
of red, blue, and green dominoes.
Clearly, a0 = 1 (s
MATH 387-01
Problem Set #2
Finding and using appropriate statistics
You can use either a standard combinatorial statistic (exponents, binomials, multinomials,
etc.), or a combination of multiple combinatorial statistics to solve these problems; explain
wh
MATH 387-01
Problem Set #1
1. (5 points) Determine how many 4-letter strings there are using the letters A, B,
and C such that each letter appears at least once in the string.
We consider all 81 possible strings in lexicographic order, only writing down t
MATH 387
Practice Exam #1
1. (12 points)
(a) (3 points) How many even four-digit numbers have at least one 7 appearing as a digit?
It is easier to count the total number of even four digit numbers, and subtract those
which have no 7s in. There are 9 10 10