Expected number of triangles in a random graph of size $n$
Set $t\stackrel{\rm def}{=}\binom{n}{3}$, fix any ordering of the $t$ possible sets of 3 nodes, and consider accordingly $X_1,\dots,X_t$ the indicator random variables where $X_j$ is equal to $1$ iff the $j$-th set defines a triangle. You are interested in $\mathbb{E}\sum_{j=1}^t X_j$, where the expectation is taken over the $\binom{n}{2}$ i.i.d. draws defining the edges.
By linearity of expectation, $$ \mathbb{E}\sum_{j=1}^t X_j = \sum_{j=1}^t \mathbb{E} X_j = t \mathbb{E} X_1 = tp^3 $$ as all $X_j$'s are identically distributed (for the second equality).