HW5
Recall that the game Set has 81 cards and 1080 ‘sets’ (which we call lines, to distinguish them
from ‘sets’ in the generic sense). You may refer to the Wikipedia page for the game as needed.
To count lines, we may apply the double-counting principle (the principle of counting elements
of a set in two different ways). Consider the set T consisting of all triples (A,B,L) where A and B
are distinct cards, and L is the line containing A and B. How many elements (i.e. triples) are in
T? On the one hand, there are 81 choices for A, then 80 choices of B (distinct from A); and then
the line L containing A and B is uniquely determined; thus |𝑇| = 81 ∗ 80 = 6480. If x is the
number of lines, then we have another way to count elements of T: there are x choices of L, and
inside L there are 3 choices of A followed by 2 choices of B (distinct from A), making altogether
|𝑇| = 𝑥 ∗ 3 ∗ 2 = 6𝑥. We must have 6480 = |𝑇| = 6𝑥; therefore 𝑥 = 1080.
Now recall that there are many affine planes of order 3 in the game Set; each such plane is an
instance of 9 cards which form 12 lines. Today’s homework question is: how many such affine
planes are there in the entire deck? (Do not list them all.)
As a hint, count in two different ways the number of ordered 4-tuples (A,B,C,P) where the cards
A,B,C form a triangle (i.e. not a line) and P is an affine plane containing A, B and C. You may
choose either to follow this hint or not.