Mathematics for Computer Science

14.7. Counting Practice: Poker Hands 567

The Binomial Theorem explains why thenchooseknumber is called abinomial

coefficient.

This reasoning about binomials extends nicely tomultinomials, which are sums

of two or more terms. For example, suppose we wanted the coefficient of

bo^2 k^2 e^3 pr

in the expansion of.bCoCkCeCpCr/^10. Each term in this expansion is a

product of 10 variables where each variable is one ofb,o,k,e,p, orr. Now, the

coefficient ofbo^2 k^2 e^3 pris the number of those terms with exactly 1b, 2o’s, 2

k’s, 3e’s, 1p, and 1r. And the number of such terms is precisely the number of

rearrangements of the word BOOKKEEPER:

10

1;2;2;3;1;1

!

D

10Š

1Š 2Š 2Š 3Š 1Š 1Š

:

This reasoning extends to a general theorem:

Theorem 14.6.5(Multinomial Theorem).For alln 2 N,

.z 1 Cz 2 C


Czm/nD

X

k 1 ;:::;km 2 N

k 1 C


CkmDn

n

k 1 ;k 2 ;:::;km

!

z 1 k^1 z 2 k^2 


zkmm:

But you’ll be better off remembering the reasoning behind the Multinomial The-

orem rather than this cumbersome formal statement.

14.7 Counting Practice: Poker Hands

Five-Card Draw is a card game in which each player is initially dealt ahandcon-

sisting of 5 cards from a deck of 52 cards.^3 The number of different hands in

(^3) There are 52 cards in a standard deck. Each card has asuitand arank. There are four suits:
(spades) ~(hearts) |(clubs) }(diamonds)
And there are 13 ranks, listed here from lowest to highest:
Ace
A; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ;
Jack
J ;
Queen
Q ;
King
K :
Thus, for example, 8 ~is the 8 of hearts andAis the ace of spades.