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## Main Question or Discussion Point

I am trying to understand a general algorithm for calculating the probability of a hand of n cards drawn from a deck containing a given set of cards. Where the given set of cards is n in size it is easy. I am struggling when the number of cards specified is less than n.

To make it easier to udnerstand I am working with a deck of 12 cards. A,2 and 3 of each suit only.

For example. From my 12 card deck I draw 4 cards.

What is the probability at least 1 is a club and at least 1 is a spade.

The probability that only 1 is a club and only 1 is spade is easy 3c1*3c1*6c2 = 135/12c4

But the answer is 258/12c4

The missing 123 are CCCS (3), SSSC (3), CCSS (9), CCSx (54) and SSCx (54) where x is not a C or S.

Is there a clever formula to count these 123 or do I need to work them out individually as I have above.

To make it easier to udnerstand I am working with a deck of 12 cards. A,2 and 3 of each suit only.

For example. From my 12 card deck I draw 4 cards.

What is the probability at least 1 is a club and at least 1 is a spade.

The probability that only 1 is a club and only 1 is spade is easy 3c1*3c1*6c2 = 135/12c4

But the answer is 258/12c4

The missing 123 are CCCS (3), SSSC (3), CCSS (9), CCSx (54) and SSCx (54) where x is not a C or S.

Is there a clever formula to count these 123 or do I need to work them out individually as I have above.