Let $B \in \text{Binomial}(N, \theta)$ be a D208: Binomial random natural number.
Let $t \in \mathbb{R}$ be a D993: Real number.

By definition, $B = \sum_{n = 1}^N X_n$ for an independent sequence $X_1, \ldots, X_N \in \text{Bernoulli}(\theta)$. Thus, applying results

(i)	R2402: Characteristic function of sum of independent random euclidean real numbers
(ii)	R3200: Characteristic function of Bernoulli random boolean number

one has \begin{equation} \begin{split} \mathbb{E}(e^{i t B}) = \prod_{n = 1}^N \mathbb{E}(e^{i t X_n}) = \prod_{n = 1}^N [\theta e^{i t} + 1 - \theta] = (\theta e^{i t} + 1 - \theta)^N \end{split} \end{equation} $\square$