2019 AMC 10 B
Complete problem set with solutions and individual problem pages
How many sequences of and of length are there that begin with a , end with a , contain no two consecutive , and contain no three consecutive ? (2019 AMC 10B Problem, Question#25)
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We can deduce, from the given restrictions, that any valid sequence of length will start with a followed by either or . Thus we can define a recursive function , where is the number of valid sequences of length . This is because for any valid sequence of length , you can append either or and the resulting sequence will still satisfy the given conditions. It is easy to find and by hand, and then by the recursive formula, we have .
After any particular , the next in the sequence must appear exactly or positions down the line. In this case, we start at position and end at position , i.e. we move a total of positions down the line. Therefore, we must add a series of and to get . There are a number of ways to do this:
Case 1: nine - there is only way to arrange them.
Case 2: two and six - there are ways to arrange them.
Case 3: four and three - there are ways to arrange them.
Case 4: six - there is only way to arrange them. Summing the four cases gives .
