Words of length $10$ in alphabet $\{a,b,c\}$ such that the letter $a$ is always doubled 
Compute the number of words of length $10$ in alphabet $\{a,b,c\}$ such that letter $a$ is always doubled (for example "$aabcbcbcaa$" is allowed but "$abcbcaabcc$" is forbidden).

I am looking for a quick/efficient way to resolve this problem. I thought of fixing "$aa$" in the beginning then draw a tree of the next possibilities but this tree will end up to be a whole forest. Can you help me ?

You can break this up into cases based on how many aa pairs there are, which can range from $0$ to $5$. Consider, for example, the case that you have $3$ aa pairs. Then you have four other letters that are all either b or c, so there are $2^4$ ways to choose the subsequence formed by those letters. Then thinking of each aa pair as a single letter, you have $7$ letters, and you need to choose $3$ of them to be an aa pair. This gives $2^4{7\choose 3}$ words with $3$ aa pairs.
In general, the number of words of this form with $i$ aa pairs is $2^{10-2i}{10-i\choose i}$. Then add up all terms of this form as $i$ goes from $0$ to $5$.
