Question is #3 in the attached pic. Please help!
$\displaystyle E(t^X)=\sum_{x=0}^n{n\choose x}t^xp^x(1-p)^{n-x}$
$\displaystyle =\sum_{x=0}^n{n\choose x}(tp)^x(1-p)^{n-x}$
Use the binomial theorem
$\displaystyle (pt+(1-p))^n$
If you differentiate and let t=1, you will obtain E(X)=np.
Differentiate a second time and let t=1 and you will find $\displaystyle E(X(X-1))$
This should work well with the Poisson too.