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I have 3 chairs to place within 8 spots. All three of my chairs come in 4 different colors.

Can I represent the number of possible placements as:

${24}\choose{3}$

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Question not clear. Can you distinguish among chairs of the same color? Do you have to use exactly one of each of the three chairs? –  Gerry Myerson Jan 16 '12 at 21:54
    
Yes, you can distinguish. Apologies. You have a red/green/blue/orange stool, a red/green/blue/orange couch, and a red/green/blue/orange crate. You MUST place exactly one stool, exactly one couch, and exactly one crate (each into one of the 8 positions). They cannot reside in the same position. –  sdasdadas Jan 16 '12 at 21:56

2 Answers 2

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OK, then choose a stool (4), a couch (4), and a crate (4), so $4\times4\times4=64$ ways to choose the furniture. Then put the stool somewhere (8), the couch somewhere (7), the crate somewhere (6), so $8\times7\times6=336$ ways to places the chosen furniture. All told, $64\times336$.

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Your suggestion is equivalent to choosing three places out of 24 - but if you have the 24 places you get by taking each of the 8 original spaces and making it into three spaces, one for each colour, you don't have a free choice - you couldn't choose the first three spaces out of the 24 new spaces, because that would represent a stack of three chairs in the first of your 8 spaces.

What you could do would be to choose the three places from the eight you are given (you clearly know how to do that), and then determine the colours of the three chairs independently. I'm sure you can figure out how that would work.

You can tell that your suggestion goes wrong because choosing from 24 places gives you a factor 23 which won't cancel, and it is quite easy to see that this could not come from the original problem in any natural way.

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Sorry, that assumed that the factor 3 came from there being three colours, but the answer still makes sense, I hope. –  Mark Bennet Jan 16 '12 at 22:00

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