Suppose we tile dominoes on a $2 \times n$ grid. We can insert one domino vertically in the first column, leaving a $2 \times n-1$ grid remaining. Or, we can insert two dominoes horizontally in the first two columns, leaving a $2 \times n-2$ grid remaining. If we let $f(n)$ be the number of ways to tile dominoes in a grid of size $2 \times n$, then the following must hold $$ f(n) = f(n-1) + f(n-2). $$ We can see that this is the Fibonacci relation. We also see that $$ f(1) = 1, $$ $$ f(2) = 2. $$ Therefore, $f(n)$ is the $(n+1)$th Fibonacci number. For a grid of size 10, we simply need to find the 11th Fibonacci number. A quick search gives us the result: 89.