COMBINATORICS AND GENERATINGFUNCTIONOLOGY: PRE-CLASS PROBLEMS
1. AMC / AIME Problems
The problems in this section are all taken from, or inspired by, AMC (American Mathematics Competition)
and AIME (American Invitational Mathematics Examination) tests from years past. They are presented in
roughly increasing order of difficulty.
The AIME consists of 15 questions to be solved in 3 hours—thus, the intent is for the following 5 problems
to be solvable in roughly an hour, assuming familiarity with basic combinatorics methods. However, the class
will cover precisely these basic methods, so don’t feel discouraged if you find yourself stumped or spending
well over an hour to solve these problems.
(1) In a standard 52-card poker deck, each card is labeled with a number from 1-13 (where “Ace” = 1,
“Jack” = 11, “Queen” = 12, and “King” = 13) and one of four suits. A pair of cards whose numbers
match is removed from such a deck, and another pair is subsequently chosen at random from the
reduced deck. What is the probability that the numbers on the second pair of cards match?
(2) Fun with regular octahedra: Six ants stand on the vertices on an 8-sided die in the shape of a
regular octahedron. Simultaneously, each begins to move along an edge towards another, adjacent
vertex, stopping when it reaches the new vertex. Assuming each ant chooses its direction randomly
and independently of all the others, what is the probability that no two ants will land on the same
vertex?
Remark 1. There are multiple ways to solve this problem, but the quickest involves thinking about
how the bugs’ motion divides the octahedron’s vertices into cycles—for example, a 2-cycle would be
a pair of vertices whose bugs simply swap places, a 3-cycle would be a triplet of vertices whose bugs
“rotate” into each others’ positions, etc. In how many ways can the vertices of the octahedron be
divided into cycles? In how many ways can a given cycle be traversed, as a function of its size?
(3) There are 19 homes on Mathematician’s Row. The neighborhood rules concerning mail delivery are
very strict:
• No two adjacent homes can receive mail on the same day;
• No more than two homes in a row are allowed to receive no mail on any given day.
How many different patterns of mail delivery are possible on any given day?
Remark 2. Consider how to set up a recursive solution to this problem—can you write down a
recursive relationship for the number of mail-delivery patterns as a function of the number of homes?
You should find that this is hard to do, but if you instead write that total as the sum of patterns
with the three different possible two-house terminations (i.e., mail - mail, no mail - mail, and mail no mail), a recursive solution becomes possible.
Remark 3. This problem can be analyzed using generating functions as well...
(4) More fun with regular octahedra: Suppose you paint the sides of a regular 8-sided die in such
a way that each side is marked with a number between 1 and 8, and each number between 1 and
8 is used exactly once. Given a randomly-generated numbering, what is the probability that no
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COMBINATORICS AND GENERATINGFUNCTIONOLOGY: PRE-CLASS PROBLEMS
two adjacent sides will be labeled with consecutive numbers (where 1 and 8 are considered to be
“consecutive”)?
(5) The revenge of the regular octahedron: Consider again the 8-sided die from the previous
problem. Given the same setup (i.e., the numbers 1-8 are randomly assigned to the sides of the
die), how many distinguishable numberings are possible? (Here, two numberings are said to be
“distinguishable” if one cannot be rotated to look like the other.)
2. Generating Functions
The problems in this section can all be solved using methods involving generating functions—but there
may be other routes to their solutions as well. Unlike the problems in the previous section, these problems
are somewhat open-ended and are not intended to be “solved” in a fixed amount of time—rather, they are
meant to motivate the fundamental ideas behind using generating functions to count things, and to provide
examples of problems in which these methods are useful.
In other words, while the problems in the previous section all admit relatively “clean,” self-contained
solutions, the problems below are more of an “exploratory” nature—they’re meant to be chewed on, mulled
over, and played with, not disposed of in one sitting.
(1) Generating functions warmup: In how many ways can I select n items from a set consisting of
k elements, assuming repetition is allowed?
Remark 4. If repetition is not allowed, the answer is given by the binomial coefficient nk . One
way to think of this is that the answer is precisely the coefficient of xn in the expansion of (1 + x)k .
Can you develop a similar interpretation for the problem as stated (i.e., with repetition) using finite
polynomials? How about infinite polynomials?
Remark 5. Just because a polynomial is infinite doesn’t mean it’s illegitimate or ill-defined—in
fact, chances are that you’ve seen examples of at least one type of infinite polynomial that can be
explicitly summed (even if you didn’t realize that’s what you were seeing!).
, and the sum of the first n squares is n(n+1)(2n+1)
.
(2) The sum of the first n natural numbers is n(n+1)
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What is the sum of the first n cubes? How about the first n fourth powers? Is there an efficient
method for computing the sum of the first n k th powers?
Remark 6. Consider an infinite polynomial whose coefficients are just the integers, i.e., 1 + x +
2x2 +3x3 +· · · . Is there a simple way of summing this infinite series? (Hint: think derivatives!) Once
you’ve done this, there is a surprisingly simple way of finding the sum of the first n coefficients—in
fact, this method generalizes to any polynomial (infinite or not)!
(3) In how many distinct ways can I write 102 in the form m + n, where m is an even positive integer
and n is a positive integer divisible by 3? Can you generalize this?
Remark 7. In the spirit of the previous remarks, there is a way of interpreting this problem in
terms of the coefficient of x102 in the expansion of a certain (infinite) polynomial. Think about how
to build such a polynomial as the product of two other infinite polynomials.