Let $X_i\stackrel{\mathrm{i.i.d.}}\sim\mathsf{Ber}(p)$ random variables for $i=1,2,\ldots$, then the moment generating function of $X_1$ is
$$
\mathbb E[e^{tX_1}] = 1-p + pe^t.
$$
Set $S_k = \sum_{i=1}^k X_i$, $k=1,2,\ldots$. Then the moment generating function of $S_k$ is given by
\begin{align}
\mathbb E[e^{tS_k}] &= \mathbb E\left[e^{t\sum_{i=1}^k X_i} \right]\\ &= \prod_{i=1}^k \mathbb E[e^{tX_i}]\\ &= \prod_{i=1}^k \mathbb E[e^{tX_1}]\\ &= \mathbb E[e^{tX_1}]^k \\
&= (1-p + pe^t)^k.
\end{align}
Set $p=\frac29$ and we have your function $M(t)$. Then
$$
\mathbb P(X=3) =\binom k 3 \left(\frac29\right)^3\left(\frac79\right)^{k-3}= \frac8{343}\binom k 3 \left(\frac79\right)^k.
$$
To determine the factorial moments of $X$, we instead use the probability generating function $P(s) = \mathbb E[s^X]$. Here $P(s) = \left(\frac79 + \frac29s\right)^k$, and the $r^{\mathrm{th}}$ factorial moment is given by
\begin{align}
\lim_{s\to1^-} \frac{\mathsf d^r}{\mathsf ds^r} P(s) &= \lim_{s\to1^-} \frac{\mathsf d^r}{\mathsf ds^r} \left(\frac79 + \frac29s\right)^k\\
&= \left(\frac29\right)^r\left(\frac79 + \frac29s\right)^{k-r}\frac{k!}{(k-r)!}, \ r\leqslant k,
\end{align}
and zero for $r>k$.