Final answer:
In response to a combinatorics problem, the student must determine the number of n-digit strings using the characters 'a', 'b', and 'c' without the substring 'aa'. This involves using a systematic approach, starting with simple cases and applying restrictions to avoid the 'aa' substring as n increases.
Step-by-step explanation:
The student is asking about a problem in combinatorics, a branch of mathematics concerned with the arrangement and number of ways certain events can occur. Specifically, the question seeks a function, f(n), that represents the number of n-digit strings made from the characters 'a', 'b', and 'c' that do not include the substring 'aa'. This is an example of a combinatorial restriction problem where one has to count the number of valid sequences under certain constraints.
To solve this, one would typically start with the case for n=1 and then build up to higher values of n by considering the possibilities at each additional character position and ensuring the 'aa' substring condition is not violated. A recursive or iterative approach can be used, taking into account both valid and invalid substrings as the string length increases.
For example, for n=1, there are three possible strings: 'a', 'b', and 'c'. For n=2, we need to consider that 'aa' is not allowed, but all other combinations of two characters are allowed, yielding a count of 'ab', 'ac', 'ba', 'bb', 'bc', 'ca', 'cb', 'cc', which is 8 possibilities. And for higher values of n, one would continue to build upon the earlier results while ensuring that 'aa' does not appear.