jmc

$43$ appears in the $43$rd row of Pascal's Triangle, since $\left(\genfrac{}{}{0ex}{}{43}{1}\right)=43$. Note that $43$ is prime, so the smallest integer $n$ such that $n!$ is divisible by $43$ is actually $43$. Therefore, the $43$rd row is the first row in which $43$ appears. Also note that all the other numbers in the $43$rd row are greater than $43$, with the exception of the $1$'s on the ends. As a result, all the numbers on the following rows will either be $1$ or greater than $43$, so $43$ only appears in the $43$rd row. Therefore, only $\boxed{1}$ row of Pascal's Triangle contains the number $43$.