Suppose a random variable X has the moment-generating function

Question

$M_X(t)=\frac{1}{7}e^{2t}+\frac{3}{7}e^{3t}+\frac{2}{7}e^{5t}+\frac{1}{7}e^{8t}$ $t\in{\rm I\!R}$
a)Convince yourself that X has discrete distribution. What is the pmf of X?
b)Find the 3rd moment of X.
For a)
You can tell that it is discrete because the MGF is the sum of $e^{tx}p(x)$ so $P[X=2]=\frac{1}{7}, P[X=3]=\frac{3}{7},P[X=5]=\frac{2}{7},P[X=8]=\frac{1}{7}$ would be the pmf?
Then for b)
the third moment can just be $M_x'''(0)$ correct? I can calculate it on my own I just wanted to be sure that I was on the right track