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Let $ a_n $ be total valid sequences of length $ n $.
Let $ b_n $ be sequences of length $ n $ ending in H, and $ c_n $ those ending in L or M. Then:
$ a_2 = 2a_1 + 2a_0 $? But $ a_0 = 1 $ (empty sequence)
$ a_3 = 2a_2 + 2a_1 = 2(8) + 2(3) = 16 + 6 = 22 $
$ a_4 = 2a_3 + 2a_2 = 2(22) + 2(8) = 44 + 16 = 60 $
$ a_5 = 2a_4 + 2a_3 = 2(60) + 2(22) = 120 + 44 = 164 $
So, there are 164 valid sequences of 5 hours with no two consecutive high readings.
Question: A virologist has 3 types of synthetic RNA sequences: 4 copies of type A, 3 of type B, and 2 of type C. If she runs them in a sequence over 9 days, one per day, how many distinct sequences can she create?
Solution: We are arranging 9 RNA segments where 4 are identical of type A, 3 of type B, and 2 of type C. The number of distinct permutations is:
Question: A primatologist records 6 social interactions among a troop, categorized as 3 grooming events, 2 play events, and 1 aggression event. If the sequence of behaviors is analyzed and events of the same type are indistinguishable, how many distinct behavioral sequences are possible?