Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 669 — #677
15.11. References 669
(c) Let H S be the set of all hands that are straights, that is, the ranks of the five
cards are consecutive. The order of the ranks is .A; 2; 3; 4; 5; 6; 7; 8; 9; 10; J; Q; K; A/;
note that A appears twice.
What is jH S j?
(d) Let H F be the set of all hands that are flushes, that is, the suits of the five cards
are identical. What is jH F j?
(e) Let H SF be the set of all straight flush hands, that is, the hand is both a straight
and a flush. What is jH SF j?
(f) Let H HC be the set of all high-card hands; that is, hands that do not include
pairs, are not straights, and are not flushes. Write a formula for jH HC j in terms of
jH NP j; jH S j; jH F j; jH SF j.
Problems for Section 15.10
Practice Problems
Problem 15.65.
Prove the following identity by algebraic manipulation and by giving a combinatorial
argument:
!
n
r
!
r
D
k
n k
!
n
r
!
k
k
Problem 15.66.
Give a combinatorial proof for this identity:
Class Problems
X
iCj CkDn
i;j;k0
!
n
i; j; k
D 3 n
Problem 15.67.
According to the Multinomial theorem, .w C x C y C z/ n can be expressed as a