Consider an $n$ tuple. Each index of the tuple is filled with a symbol chosen from the following set comprising of four symbols: \begin{equation} S = \{a, b, c, d\}. \end{equation}
Note that there are $4^{n}$ possible fillings.
I want to count the number of possible tuples such that $b$-s, $c$-s, and $d$-s all occur in it an even number of times (not necessarily the same even number for each.)
For example, if $n = 4$, then $bbba$ is not a valid tuple --- as it has $3$ $b$-s, which is an odd number. However $bb cc$ is a valid tuple — as both $b$ and $c$ occur an even number of times.
For $n = 2$, the valid choices are $aa, bb, cc, dd$.
Hence, the number of such tuples is $4$.
For $n = 3$, the valid choices are $aaa, abb, acc, add, bba, cca, dda, bab, cac, dad$.
So, $10$ choices.
Is there a general pattern?