So we're trying to distribute 5 cards over 3 holes. Here're the holes:

|_|_|_|
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Let's put the cards there. Well, they won't fit, because the '_' (underscore) occupies the whole cell, rather than just the bottom of it. Let's get rid of underscores. They're just for visials anyway.

| | | |
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Now it fits. One way to do it:

   1 1
|1|1|1|
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or maybe like this:

   1
   1
|1|1|1|
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First, the bottom row of ones is always the same. We don't actually need it to count combination. Lets just remove it (like in tetris). The number of ways to distribute 5 cards over 3 holes with BUK's constraint is the same as the number of ways to distribute 2 cards over 3 holes without this constraint.

1) |1|1| |

2) | |1|1|

        1
3) | | |1|

4) and so on
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That actually looks like a datastructure we can work with. The problem is it's two dimensional. Which is kind of a pain. But! we can flatten it. Like so:

1) | |1|1|

2) |1|1| |

3) |11| | |

4) | |11| |

5) and so on
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Now there are two more things. The spaces in empty holes are not necessary. Also, the two '|' (pipes) at the left and right edges of our strings are not necessary either. That's also just visuals. Let's remove that stuff.

1) | | |11| => ||11 => 0 0 2

2) |1|1| |  => 1|1| => 1 1 0

3) | |11| | => |11| => 0 2 0

4) etc
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For no particular reason I used '0' instead of '|' in hacky solution :) (hm, I think that's because it reminded me of a recent thread). Like that:

1) 0011 => 0 0 2

2) 1010 => 1 1 0

3) 0110 => 0 2 0
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Now we just want to count all combinations of such strings. The length of a string is $cards (which is the number of ones) + $holes - 1 (which is the number of zeros). That's our $n. And $r is obviously the number of cards. Think of it this way:

0000 - start with string of all zeros
.00. - take some zeroes that way
1001 - and replace with ones

0..0 - take some zeroes that way
0110 - and replaces with ones

00.. - then that way
0011 - you got the idea
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Hopefully now its clear that the number of ways we can produce such strings is n! / (r!(n-r)!), just like in a textbook.

And my hacky solution literally builds these strings and counts the length of sequences of ones :) Which is a bit of a nonsense, and GrandFather's method is a lot better, but hey! it works ;)