  For given integers $k$ and $\ell$ with $0<\ell< {k \choose 2}$, Alon, Hefetz,
Krivelevich and Tyomkyn formulated the following conjecture: When sampling a
$k$-vertex subset uniformly at random from a very large graph $G$, then the
probability to have exactly $\ell$ edges within the sampled $k$-vertex subset
is at most $e^{-1}+o_k(1)$. This conjecture was proved in the case
$\Omega(k)\leq \ell\leq {k \choose 2}-\Omega(k)$ by Kwan, Sudakov and Tran. In
this paper, we complete the proof of the conjecture by resolving the remaining
cases. We furthermore give nearly tight upper bounds for the probability
described above in the case $\omega(1)\leq \ell\leq o(k)$. We also extend some
of our results to hypergraphs with bounded edge size.
