Combinatorics Through Guided Discovery, 2004a

186 D. Hints to Selected Problems
1=5c − 3d.
218. Tohave
we must have
ax + b
(x − r 1 )(x − r 2 ) = c
x − r 1
+
d
x − r 2
cx − r 2 c + dx − r 1 d = ax + b.
221. You can save yourself a tremendous amount of frustrating algebra if you
arbitrarily choose one of the solutions and call it r 1 and call the other solution
r 2 and solve the problem using these algebraic symbols in place of the actual
roots.1 Not only will you save yourself some work, but you will get a formula
you could use in other problems. When you are done, substitute in the actual
values of the solutions and simplify.
222.a. Once again it will save a lot of tedious algebra if you use the symbols r 1 and
r 2 for the solutions as in Problem 221 and substitute the actual values of the
solutions once you have a formula for a n in terms of r 1 and r 2 .
222.d. Think about how the binomial theorem might help you.
224.a. A Catalan path could touch the x-axis several times before it reaches (2n, 0).
Its first touch can be any point (2i, 0) between (2, 0) and (2n, 0). For the path
to touch first at (2i, 0), the path must start with an upstep and then proceed as
a Dyck path from (1, 1) to (2i −1, 1). From there it must take a downstep. Can
you see a bijection between such Dyck paths and Catalan paths of a certain
kind?
224.b. Does the right-hand side of the recurrence remind you of some products you
have worked with?
224.c.
1 · 3 · 5 ···(2i − 3)
i!
=
(2i − 2)!
(i − 1)!2 i i! .
226. Try drawing a Venn Diagram.
228. Try drawing a Venn Diagram.
231.b. For each student, how big is the set of backpack distributions in which that
student gets the correct backpack? It might be a good idea to first consider
cases with n =3, 4, and 5.
Additional Hint: For each pair of students (say Mary and Jim, for example)
how big is the set of backpack distributions in which the students in this pair
get the correct backpack. What does the question have to do with unions or
1We use the words roots and solutions interchangeably.