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A common misconception (I think) is that if you've got a set of permuatuations (which are open to variation that could mean that they're not all distinct) the simple way to find out how many there are (ignoring what they are) is to take the number of permuations P (where variation are classed as different) and divide by the number of variations V :these could be "rotations and reflections" (for shapes) cyclic combinations of colours (for couloured patturns etc)
Why did the chicken cross the Mobius Band?
To get to the other ...um...!!!
Re: combination misconception
Maybe Burnside's Lemma will help for some of these examples.
Imagine for a moment that even an earthworm may possess a love of self and a love of others.