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Imagine you have a sequence of cards, each having a unique set of features.

Example features : letter (A, B, C), number (1, 2, 3), and color (Red, Green, Blue)

Some example cards : A1Red, B1Blue, C3Green, B2Red, etc.

I want to find patterns made of 3 cards. The rules are that for these 3 cards to "match", within a single feature, all cards must be different, or the same.

For example, just looking at the "letter" feature. ABC is a match, BBB is a match, and AAC, BAA, etc. are not.

So for a whole set example: 

A1Red, B2Green, C3Blue = match
B3Green, B3Green, B3Green, = match
C2Red, C2Blue, C2Green = match
B1Red, C1Green, A1Blue = match

A1Red, A2Blue, A1Green = NOT match (two 1s)
B2Green, A1Red, C3Red = NOT match (two reds)

So if I lay out a random 16 cards, how many matches can I expect to find in there? More generically, how many matches can I expect to find in X cards?

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  • $\begingroup$ To clarify: A single feature matches if they are all identical or all distinct, and we require all three features to match. $\endgroup$
    – Graham Kemp
    Commented Mar 26, 2014 at 23:34

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Suggestion for further research: Are you familiar with the card game Set? What you're describing is that game, restricted to three characteristics per card (the regular game has four characteristics per card). There have been many things written about that game.

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