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.

${50 \choose k} x^k x^{k-50}$.
You need $k+(k-50)=18$, therefore $2k-50=18$, and $k=34$.
Calculating ${50 \choose 34}=4923689695575$ - your answer
