DISCRETE MATH How many ways can a gambler draw 5 cards from

DISCRETE MATH

How many ways can a gambler draw 5 cards from a standard deck and get:

A) A flush (5 cards of the same suit)?
B) Four aces?
C) Four of a kind?
D) Three aces and two jacks?
E) Three aces and a pair?
F) A full house (three of a kind and a pair)?
G) Three of a kind?
H) Two pairs?

Solution

In a deck of cards there are 4 types of 13 cards each totalling 52.

A) A flush (5 cards of the same suit) = 4(13C4) =4(715) = 2860

B) Four aces = 4C4 ( 48C1) = 192

C) Four of a kind = 4C4 ( 48C1)13 = 2496

D) Three aces and two jacks

= 4C3 (4C2) = 24

E) Three aces and a pair

= 4C3(4C2) 12 (since A already counted)

= 288

F) A full house (three of a kind and a pair)

= 13(4C3) 12(4c2)

= 52(72)

= 3744

G) 3 of a kind

= 13(4C3) * 48 C2

= 52(24)(47)

=58656

H) Two pairs

= 13C2 ( 4C2) 48C1

= 78(12)48

= 44928