Classic Counting Problems Does anyone have some good, classic, counting problems? I want things that are interesting, as well as instructive- more than just compute the number of way to get a flush, etc. (Not that those aren't interesting, but students seem to get bored of that quickly.)

One way to improve the poker hand counting problem to a more interesting version is:

For a time in the 19th century there was a vogue for 65-card decks with five suits of 13 cards each.  Calculate the proper ranking of poker hands when playing with such a deck.
In particular, does a full house still beat a flush?

Or similarly:

On the planet Pyrzxqgl, the inhabitants play poker with a standard 52-card deck, but the hands have 6 cards each.  Calculate the proper ranking of poker hands on the planet Pyrzxqgl.  Don't forget to include a ranking for three pairs! 

Or have them invent a new poker hand (with the standard deck) such as:


*

*a four-card straight plus a pair (for example, 4 5 5 6 7)

*all cards the same color (for example, 3♠ 9♠ J♠ and 7♣ Q♣)

*a pair, with the three odd cards all the same suit (for example, 2♡ 2♣ 5♠ 6♠ Q♠) 

*no card higher than a 7

*five face cards (this is semi-standard; it is called a blaze).


and have them rank their new hand relative to the standard hands.  
Or turn the problem around: The odds off getting a full house are much better than the odds of getting the next higher-ranked hand, four of a kind.  Invent a poker hand that should rank as in between the two, as near to the middle as possible.  (Bonus question: what does "near to the middle" mean here?)
