How many ‘words’?
November 4, 2006 pm30 11:39 pm
How many 3-letter words can be made using the ‘alphabet’: { :) , :( , X , O} if :) and :( are not allowed to appear in the same word?
Examples of 3-letters words: :) O O , X :( O
:) X :( is a 3-letter word that we are not counting.
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JD2718 
Lessee … Your alphabet is basically { face, X, O } and there are two choices for face: happy and sad. The rule about not having both a happy and a sad face in the same word can be expressed as “whatever you choose for face, it’s the same for that whole word”. So the answer is two times the number of combinations of { face, X, O }, or 2*(3^3) = 54.
Thinking of your last post, I asked google “two times three cubed” and it says “fifty-four”. Cute.
But some of those words (XXO, for example) are in both sets of 27, and shouldn’t be counted twice, so isn’t it 2*(3^3)-2^3=46?
46 it is. The other approach is subtraction: there are 4^3 = 64 total strings of those 4 symbols, of which 3 are permutations of [ :) :) :( ], 3 are permutations of [ :) :( :( ], 6 are permutations of [ :) :( O ] and 6 are permutations of [ :) :( X ]. That leaves 64 – 18 = 46 words.
Ah, well reasoned both of you. This is such a cool blog not only because of the good posts but also the good commenters!
Thanks for the kind words, mrc. My apologies for disappearing these last couple of weeks.
My solution was identical to rdt’s, but JBL’s is absolutely equivalent.
It is good for my students to see that not every combinatorial problem can be answered with 15, 24, 32 or C(n,r). I like how ‘non-special’ 46 is.