Determine the coefficient of $x^{18}$ in $\left(x+\frac{1}{x}\right)^{50}$ 
Determine the coefficient of $x^{18}$ in $\left(x+\frac{1}{x}\right)^{50}$.

I know he Binomial Theorem will be useful here, but I am struggling to use it with any certainty.

Since for $y=x^2$,
$$\left(x+\frac{1}{x}\right)^{50}=\frac{\left(x^2+1\right)^{50}}{x^{50}}=\frac{\left(y+1\right)^{50}}{y^{25}}$$
you are looking for the coefficient of $y^{9+25}=y^{34}$ in $(y+1)^{50}$.
