  The recent observations of supermassive black holes (SMBHs) at high redshift
challenge our understanding of their formation and growth. There are different
proposed pathways to form black hole (BH) seeds, such as the remnants of the
first stars (chapter 4), gas-dynamical processes (chapter 5), direct collapse
(chapter 6), or stellar collisions in dense nuclear clusters (chapter 7). In
this chapter, we discuss the probability of forming supermassive black holes
(SMBHs) via these channels and determine the expected number density of the BH
seeds. We start with a brief discussion of the observational constraints on
SMBHs at low and high redshift that theoretical models have to reproduce (a
more detailed account is provided in chapter 12). We further present the most
popular formation channels of SMBHs, discuss under which conditions they can
reproduce the observations, and compare various estimates in the literature on
the expected number density of SMBHs. To account for the density of quasars at
$z>6$ requires very efficient gas accretion mechanisms or high BH seeds masses.
The bottleneck to obtain sufficiently high number densities of seed BHs with
masses $>10^5$M$_\odot$ is the interplay between radiative and chemical
feedback, which constrains the conditions for primordial, isothermal gas
collapse.
