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Current time:0:00Total duration:2:58

2003 AIME II problem 3

Video transcript

define a good word is a sequence of letters that content that consists only of the letters a B and C some of these letters may not appear in the sequence and in which a is never immediately followed by B B is never immediately followed by C and C is never immediately followed by a how many seven letter good words are there so let's just think about this a little bit so there's letters with just a is B's and C's and then it could be all A's all B's all C's so that's up because some letters might not appear and a is never immediately followed by B so a can only be followed by another a or another C B is never immediately followed by C which means that B can only be followed by an A or another B and C is never immediately followed by a so C can only be followed by another C or a B so how many seven letter good words are there so let's just think about the places we have seven letters so one two three four five six seven letters now there's no constraints on this first letter since it's not following anything so it could be an a B or a C so there's three there's three possibilities for this first letter now there's three possibilities for this first letter but no matter what pot what no matter what letter this is how many possibilities are there for this second letter over here well if this was an A you could the second letter could only be an A or a C because it can't be followed by B if this was a B it could own the second letter could only be a B or an A because it can't be followed by a see if this was a C the second letter could only be a B or C so no matter what letter this first letter is the second letter can only have two possibilities there could only be two possibilities or another way to think about it is there's one letter no matter what letter this is there's one letter that's being ruled out here so it can only be two possibilities well the same thing is here we're going to stick some letter here and no matter what letter there is over here it's going to rule out one possibility over here so we're going to have only two possible letters that we can put here no matter what letter is there and use the same logic only two possibilities there only two possibilities there only two possibilities there and then only two possibilities there so how many total possibilities do we have well 3 times 2 times 2 times 2 times 2 times 2 this is 1 2 3 4 5 6 2's so this is equal to 3 times 2 to the 6th power which is 3 times 2 to the 2 to the 6th is 32 times 2 is 64 which is equal to 180 plus 12 is equal to 192 there's 192 possible good 7 letter good words or good words is defined above