Question

Consider a reduced deck where all cards 2, 3, 4, and 5 (16 cards) are removed. In a game of five card draw find the number of ways to draw one pair, two pair, three of a kind, straight, flush, full house, four of a kind, straight flush, and royal flush.

Answer 1

one pair

choose one card out of 9 then decide any 2 from the 4 suits 9C1 4c2

rest two places can have any of the others 8C3 (4c1)^3

so total = 9c1 4c2 8c3 (4c1)^3

9*6*56* *4^3= 193536

b)

two pairs

choose two values for two pairs and decide suits 9c2 4c2 4c2

choosing last card 8C1 4c1

9c2 4c2 4c2 8C1 4c1 = 36*6*6*8*4 = 41472

c)

three of a kind

choose one card and decide its suit 9c1 4c3

choose two other cards and dcide their suit 8c2 4c1 4c1

9c1 4c3 8c2 4c1 4c1

= 9*4*28*4*4 =16128

d)

full house

three cards of one value tow of other

4c1 4c3 3c1 4c2 =4*3*3*6 = 216

e)

four of a kind

choose a card and all suit

9c1 4c4

choose another card of one suit 8c1 4c1

9c1 4c4 8c1 4c1 =9*1*8*4 = 288

flush,

4c1 9c5 - 20 = 504 -20 = 484

we subtact the nu,be rof royal and straight flushes

straight,

all cards in a increasing sequence

lowest card can be A, 6, 7, 8, 9 ,10

then choose the suit of each

6*(4c1)^5= 6*4^5 =6144

straight flush

we have a staight with lowest card being 6,7,8,9

and 4 suits

so 4*4 = 16

royal flush

we only can choose a suit

4c1 =4