In a game of bridge, each player gets 13 cards. In how many ways can a person get a bridge hand consisting of only aces or face cards?

Expert Answers
embizze eNotes educator| Certified Educator

There are 4 aces, 4 kings, 4 queens, and 4 jacks in a standard deck. So the number of aces and face cards is 16.

The number of hands that contain only aces and face cards is the number of 13 card hands that can be created from these 16 cards, or `_16C_13=560` .

Another way to think of this is to consider the number of ways of choosing the 13 cards from the 16 honors: `(16!)/(3!)~~3.487131648x10^(12)` (There are 16 choices for the first card, 15 choices for the second, etc...) But the order of the cards is not important, so we divide by the number of different ways we could deal the same cards which is `13!` , resulting in 560. This is, of course, the definition of the combination used above. `(16!)/((3!)(13!))`