Given m blue identical marbles and n red identical marbles. How many sequences of these m+n marbles are there so that no two blue marbles are next to each other?
Attempt: The Fibonacci sequence does not solve this, since there are a fixed number of each color marbles.
Changing terminology to $r$ for red marbles, $b$ for blue marbles for better correspondence,
we need $b \le (r+1)$ if the non-adjacency conditions are to be met.
There are $(r+1)$ places (uparrows) where the blue marbles can be placed
$\uparrow\color{red}\bullet\uparrow\color{red}\bullet\uparrow\color{red}\bullet\uparrow ....$
Thus the $b$ blue marbles can be placed in $\binom{r+1}{b}$ non-adjacent ways