In the following question I am trying to determine how many DNA sequences of length $3$ that have no $C$'s at all or have no T's in the first position.
Below are my workings,
So there are $4$ DNA letters, $A,T,C,G$
Considering how many DNA sequences of length $3$ that have no $C'$s,
in the first position we have $3$ options, then in the next position we have another $3$ options to fill in, and then finally in the last position since its length $3$ we have $3$ options, therefore by the product rule we have,
$$3x3x3 = 3^3 = 27$$
Considering DNA sequences of length $3$ that have no $T's$ in the first position
in the first position we have $3$ options, then in the next position we have $4$ options to fill in, finally in the last position since its length $3$ we have $4$ options, therefore by the product rule,
$$3*4*4 = 3*4^2 = 48$$
So my questions are the following,
1) Am I using the product rule correctly?
2) What does the or mean in this case? Is it two different questions or is it all one question and I will need to use the sum rule to combine both answers above?