Exam #2 — Math 173 — Spring ’09
Practice Problems
(1) Find a generating function that generates the sequence 2,0,2,0,2,0, . . . (i.e., a0= 2, a1= 0,
a2= 2, a3= 0,. . . .
(2) Find a generating function for the number of ways to distribute 12 indistinguishable balls
into 4 indistinguishable boxes.
(3) Give the ordinary generating function for the sequence 2,5,13,35, . . . = 21+ 31,22+32,23+
33, . . . in closed form.
(4) Compute [x12](1 −4x)−5
(5) For each r≥4, let ardenote the number of solutions to the equation p+q=rfor which p
and qare both primes. Let f(x) = x2+x3+x5+x7+x11 +· · · be the generating function for
the prime numbers. Using f(x), find the generating function for the sequence a0, a1, a2, . . ..
(FYI: The Goldbach Conjecture proposes that ar≥1 whenever ris an even number
greater than 2; i.e., every even number can be written as the sum of two primes. This is a
famous unsolved problem in number theory.)
(6) MacGyver has pinpointed three files as the source of potential problems on a computer. He
has used each file in a test 12 times, each pair of files in the same test together 6 times, and
all 3 files in the same test together 4 times. In 8 tests, none of the files were used. How
many tests were performed altogether?
(7) How many integers between 1 and 600 (inclusive!) are divisible by none of 2, 3 and 5?
(8) Find a generating function for the number of nonnegative integers solutions to the equation
x1+x2+x3= 20 in which x1is odd, 2 ≤x2≤5 and x3is prime. Do not try to compute
the numerical answer.
(9) Write down the partitions of 12 into distinct even parts.
(10) Show that the number of partitions of ninto three parts equals the number of partitions of
2ninto three parts of size strictly less than n. (Hint: draw the Ferrers diagrams for some
small cases such as n= 5 and n= 6 and argue in terms of pictures.) Note: This one is a
little tricky which is the main reason it didn’t make it on the exam; nice problem, though.
(11) In making an 11-problem exam, an instructor want to use at least 3 easy problems, at
least three medium problems, and at least two hard problems. She has available 7 easy, 6
medium and 4 hard. In how many ways can she make the test? (Do not distinguish two
different problems of the same difficulty from each other.)
(12) How many different committees of 40 US senators can be formed if the two senators from
the same state are considered identical?
(13) A survey is done of 200,000 people. According to the report, 130,000 are males, 90,000 are
smokers and 10,000 have cancer. However, of those surveyed, there are 7,000 males with
cancer, 8,000 smokers with cancer, and 5,000 male smokers. Finally, there are 1,000 male
smokers with cancer. How many female nonsmokers without cancer are there? There is a
problem with the report. How can you tell?
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