How many combinations of five card hands are there in a 52 card deck
where three will be within hearts and two within clubs, as well how
can one ascertain the probability of this
outcome?

H = heart, C = club

the possible combinations of success are:
HHHCC
HHCHC
HHCCH
HCHCH
HCHHC
HCCHH
CHCHH
CHHCH
CHHHC
CCHHH

That is 10 possible combinations out of a 1024 possibilities (4^5, 4
suits to the fifth power --because you draw 5 cards--).

So the chances are 10/1024 or .9765625%.

I'm 75% sure that this is the right answer... anyone is welcome to
give a better answer if you can explain why I'm wrong :)

I'm not entirely sure about this either but, I beleive that the total
number of hands in a 52 card deck is:

52!/(5!47!) = 2598960   (aka 52 choose 5)

The total number of hands with 3 hearts and 2 clubs can be found as follows:

There are 13 hearts in a deck so there will be 13!/(3!10!) = 256 (13 choose 3)
possibilities of choosing the 3 hearts. And there are 13 clubs in the
deck so there will be 13!/(2!11!) = 78 (13 choose 2) possibilities for
the clubs.

Multiplying the two together:

(13 choose 3)(13 choose 2) = 286*78 = 22308 possible hands with 3
hearts and 2 clubs

Finally the odds of getting this hand will be 22308/2598960 or 0.858%

Here is some support for Rrandomeh's approace:

13 choose 5 = 1287

So according to your logic there would be a 1287/2598960 chance of
drawing 5 hearts.... or 0.000495198079

And according to http://wizardofodds.com/games/pokerodd.html, the odds
of getting a flush are 0.00196540
Which when divided by 4 (to give the odds of drawing a flush of
hearts) is .00049135 which is surprisingly close to the number above
(.000495198079).

So Rondomeh must be on to the correct method here and I wish to erase
my answer above to save face around here :)

I would love to be a Google Researcher.  Until that time, here is your answer:

Your question:
"How many combinations of five card hands are there in a 52 card deck
where three will be within hearts and two within clubs, as well how
can one ascertain the probability of this outcome?"

The simplest way to count these is to realize that there are two separate elements:
1 - Number of clubs/hearts
2 - Number of different permutations of any given group of clubs/hearts

The different combinations of any hand involving (3) hearts and (2) clubs is:

HHHCC	HHCHC	HCHHC	CHHHC	HHCCH
HCHCH	CHHCH	HCCHH	CHCHH	CCHHH

However, there are 13 possible cards from each suit.  Each combination
must take this into consideration.

For HHHCC, there are (13)(12)(11)(13)(12) possible permutations, or
267,696 different ways that you could have three hearts then two
clubs.  Multiply this times the ten different combinations, and you
get 2,676,960 different combinations of five card hands where three
will be within hearts and two with clubs

so the answer to your first question is 2,676,960 different combinations

the second answer requires simply knowing the number of different
hands possible in a deck of cards.

The total possible hands is (52)(51)(50)(49)(48) or 311,875,200

To calculate any probability, you simply divide the number of desired
outcomes by the number of possible outcomes.

So, P(3 hearts + 2 clubs) = 2,676,960 / 311,875,200
or 0.8583 percent (less than 1 percent)