Question

You have a deck of 52 cards, and you draw the top 7 cards.
a) How many hands could you get?
b) How many hands will have 4 of a kind?
c) How many hands will have 5 of the same suit?
d) How many hands will have a 7 card flush?

Answer 1

a) No of cards available, n = 52, cards to be chosen, r = 7

No of ways of selection = C(n,r) = 52!/(45!7!) = 133784560

b) For of a kind: 4 cards of same rank. Since there are 13 different ranks and each rank has four suits. So no of ways of choosing those 4 cards = 13. Now for remaining 3 cards, we have 48 choices.

So no of ways = 13•C(48,3) = 13•(48!/(45!3!)) = 224848

c) Choose any one suit if of 4, them choose 5 cards out of 13 from that suit. For remaining 2 cards, there are 39 cards available. So no of ways = 4•C(13,5)•C(39,2) = 3814668

d) “7 Card Flush” is a hand consisting of seven cards of the same suit, regardless of rank.

So, choose any suit (4 ways) then choose 7 cards out of 13 from that suit.

No of ways = 4•C(13,7) = 6864