Clique count in a binomial random graph
= Clique count in a binomial random graph
If $X_t$ counts copies of the <complete graph> $K_t$ in $G(n,p)$, then
$$
\mathbb EX_t=\binom ntp^{\binom t2}.
$$
Consequently $p n^{2/(t-1)}\to0$ implies $\mathbb P(X_t>0)\to0$ by the <first moment method>.