# how many sequences of ‘s and ‘s of length are there that begin with a , end with a , contain no two consecutive ‘s, and contain no

how many sequences of ‘s and ‘s of length are there that begin with a , end with a , contain no two consecutive ‘s, and contain no three consecutive ‘s?

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1. After any particular a, the next a in the sequence must appear exactly 2 or 3 positions down the line. In this case, we start at position b and end at position 19. answer is 65
i.e. we move a total of 18 positions down the line. Therefore, we must add a series of c’s and d’s to get 18. There are several ways to do this:
Case 1: nine c’s – there is only 1 way to arrange them.
Case 2: two d’s and six c’s – there are 28 ways to arrange them.
Case 3: four d’s and three c’s – there are 35 ways to arrange them.
Case 4: six c’s – there is only 1 way to arrange them.
Summing the four cases gives 1+28+35+1 = 65