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?

a. 108336, b. 480200

Step-by-step 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