Combinatorics Through Guided Discovery, 2004a

187
intersections of sets. Keep on increasing the number of students for which
you ask this kind of question.
232. Try induction.
Additional Hint: We can apply the formula of Problem 226 to get
( n⋃ n−1
) ⋃
A i =
A i ∪ A n
i=1 i=1
( n−1
⋃ n−1
) ⋃
=
A i + |A n |−
A i ∩ A n
i=1
i=1
n−1
⋃ ⋃n−1
=
A i + |A n |−
A i ∩ A n
i=1
233.b. Let T be the set of all i such that x ∈ A i . In terms of x, what is different about
the i in T and those not in T?
Additional Hint: You may come to a point where the binomial theorem
would be helpful.
235. Notice that it is straightforward to figure out how many ways we may pass
out the apples so that child i gets five or more apples: give five apples to
child i and then pass out the remaining apples however you choose. And if
we want to figure out how many ways we may pass out the apples so that a
given set C of children each get five or more apples, we give five to each child
in C and then pass out the remaining k − 5|C| apples however we choose.
236. Start with two questions that can apply to any inclusion-exclusion problem.
Do you think you would be better off trying to compute the size of a union
of sets or the size of a complement of a union of sets? What kinds of sets
(that are conceivably of use to you) is it easy to compute the size of? (The
second question can be interpreted in different ways, and for each way of
interpreting it, the answer may help you see something you can use in solving
the problem.)
Additional Hint: Suppose we have a set S of couples whom we want to seat
side by side. We can think of lining up |S| couples and 2n − 2|S| individual
people in a circle. In how many ways can we arrange this many items in a
circle?
237. Reason somewhat as you did in Problem 236, noting that if the set of couples
who do sit side-by-side is nonempty, then the sex of the person at each place
at the table is determined once we seat one couple in that set.
Additional Hint: Think in terms of the sets A i of arrangements of people in
which couple i sits side-by-side. What does the union of the sets A i have to
do with the problem?
i=1