Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 637 — #645
15.11. References 637
n-element subsets.
From another perspective, the number of hands with exactly r red cards is
! !
n 2n
r n r
since there are n
r ways to choose the r red cards and
2n
n r
ways to choose the
n r black cards. Since the number of red cards can be anywhere from 0 to n, the
total number of n-card hands is:
! !
nX n 2n
jSj D
:
r n r
rD0
Equating these two expressions for jSj proves the theorem.
Finding a Combinatorial Proof
Combinatorial proofs are almost magical. Theorem 15.10.2 looks pretty scary, but
we proved it without any algebraic manipulations at all. The key to constructing a
combinatorial proof is choosing the set S properly, which can be tricky. Generally,
the simpler side of the equation should provide some guidance. For example, the
3n right side of Theorem 15.10.2 is
n , which suggests that it will be helpful to
choose S to be all n-element subsets of some 3n-element set.
15.11 References
[5], [15]
Problems for Section 15.2
Practice Problems
Problem 15.1.
Alice is thinking of a number between 1 and 1000.
What is the least number of yes/no questions you could ask her and be guaranteed
to discover what it is? (Alice always answers truthfully.)
(a)