  We prove a quantitative local limit theorem for the number of descents in a
random permutation. Our proof uses a conditioning argument and is based on
bounding the characteristic function $\phi(t)$ of the number of descents. We
also establish a central limit theorem for the number of 3-term arithmetic
progressions (3-APs) in a random subset of $\mathbb{Z}/p\mathbb{Z}$. We
conjecture that there is no local limit theorem for 3-APs, but a proof of this
remains elusive. A promising avenue of proof is to condition on the size of the
subset and show that the resulting distributions are too far apart for
different sizes. This has proven difficult because the distances between these
conditioned distributions on are the same order as their standard deviations
such that the constant multiple between them is not very large.
