Question

A standard deck of cards contains 52 cards. The cards have 4 different suits: clubs, diamonds, hearts, and spades. For each suit, there are 13 cards: one for each of the values 2 through 10, jack (J), queen (Q), king (K), and ace (A). A poker hand consists of 5 cards chosen at random from a standard deck. a. How many poker hands have two or more aces? b. How many poker hands contain the king of diamonds, the queen of hearts, or both?

Answer 1

a. 108336, b. 480200

Explanation:

a. No. of ace cards = 4

No. of non ace cards= 48

We draw 5 cards

So,

No. of ways of 2 aces = C(4,2) C(48,3)

No. of ways of 3 aces = C(4,3) C(48,2)

No. of ways of 4 aces = C(4,4) C(48,1)

Hence

No. of ways of 2 or more aces

= C(4,2) + C(48,3) + C(48,3) + C(4,3)C(48,2) +C(4,4)C(48,1)

= 6(17296) + 4(1128) + (48)

= 108336

b. No of ways any 5 cards can be chosen = C(52,5)

No. of ways kings of diamonds or queen of hearts is chosen = C(50,5)

Hence

No. of ways of choosing king of diamonds, queen of hearts or both = C(52,5) – C(50,5)

= 480200