Combinatorics/Permutation Problem; Due Friday

[shaehl]

I just can't seem to get this problem, and time is running out.

The problems on this page all deal with hands taken from a deck of 32 cards:
the 7, 8, ... , Q, K, A of each of the four suits.

How many five-card hands are possible which satisfy:
the hand has exactly two hearts
the hand has exactly two spades
the hand has exactly two Kings

For this problem I set up the three cases like this: (O = non-heart, non-spade suits)

1KH - 7H - 1KS - 7S - 14O
1KH - 7H - 7S - 6S - 1KO
7H - 6H - 1KS - 7S - 1KO

Then I multiplied and added them together and got the correct answer of 1274. However, I can't seem to figure out the second and third problems.

How many 6-card hands are possible which satisfy:
the hand has exactly two hearts
the hand has exactly three spades
the hand has exactly 2 Kings

I tried setting up the cases the same as in problem #1, with the necessary modifications:

1KH - 7H - 1KS - 7S - 6S - 14O
1KH - 7H - 7S - 6S - 5S - 1KO
7H - 6H - 1KS - 7S - 6S - 1KO

But My answer of 7350 is wrong and I can't figure out what I'm doing wrong.

Similarly, I can't figure out the 3rd problem:

How many 6-card hands are possible which satisfy:
the hand has exactly two hearts
the hand has exactly three spades

I would think it would simply be:

8H - 7H - 8S - 7S - 6S - 16O multiplied together, but it isn't.

Any Help would be great.
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[pka]

How many five-card hands are possible which satisfy:
the hand has exactly two hearts; the hand has exactly two spades; the hand has exactly two Kings
1KH - 7H - 1KS - 7S - 14O
1KH - 7H - 7S - 6S - 1KO
7H - 6H - 1KS - 7S - 1KO
Then I multiplied and added them together and got the correct answer of 1274.[/quote]
I disagree with that as being the correct answer.
In counting 1KH - 7H - 7S - 6S - 1KO you have over counted the spades.
In other words: you may have picked 7 then an 8 or 8 then a 7. But it is the same hand. The correct answer is
\(\displaystyle (1)(7)\left( {\begin{array}{c} 7 \\ 2 \\ \end{array}} \right)\left( 1 \right) = 147\).

So the total is 980 not 1274.