1. The problem is to find the moment generating function (MGF) of the binomial probability distribution. 2. The binomial distribution models the number of successes in $n$ independent Bernoulli trials, each with success probability $p$. 3. The probability mass function (PMF) of a binomial random variable $X$ is: $$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$$ where $k = 0, 1, 2, \ldots, n$. 4. The moment generating function $M_X(t)$ is defined as: $$M_X(t) = E[e^{tX}] = \sum_{k=0}^n e^{tk} P(X=k)$$ 5. Substitute the PMF into the MGF: $$M_X(t) = \sum_{k=0}^n e^{tk} \binom{n}{k} p^k (1-p)^{n-k}$$ 6. Factor the sum recognizing it as a binomial expansion: $$M_X(t) = \sum_{k=0}^n \binom{n}{k} (p e^t)^k (1-p)^{n-k} = (p e^t + 1 - p)^n$$ 7. Therefore, the moment generating function of a binomial random variable $X$ with parameters $n$ and $p$ is: $$\boxed{M_X(t) = (1 - p + p e^t)^n}$$ This function can be used to find moments of the distribution by differentiating with respect to $t$ and evaluating at $t=0$.